Pc Kevin Duffy, 52,(right) and PCSO Andrew Passmore, 56, (left) were found guilty of misconduct in a public office in connection with the death of in Bristol in 2013.
Mr Ebrahimi, 44, was beaten to death and his body set on fire by neighbour Lee James, who wrongly believed he was a paedophile.
Bristol Crown Court heard that Duffy saw Mr Ebrahimi as a liar and a nuisance despite the dangerous situation he was in.
The officer refused to speak to the Iranian refugee despite his repeated pleas in the days leading up to the murder.
Passmore was found to have lied by telling murder detectives he had patrolled outside Mr Ebrahimi’s home for an hour when it had actually been a few minutes.
Avon and Somerset Police dismissed Duffy and Passmore following misconduct hearings in January.
Judge Neil Ford QC, the Recorder of Bristol, jailed Duffy for 10 months and Passmore to four months.
During their seven-week trial, jurors heard how Mr Ebrahimi dialled 999 to report that James had come into his flat and headbutted him on July 11.
James wrongly believed that Mr Ebrahimi had filmed his young children.
He had actually been gathering evidence of anti-social behaviour.
Police arrived at the scene, Capgrave Crescent in Brislington, to find James crying with anger and frothing at the mouth.
A mob had formed outside and James was heard shouting: “Paedo! I’m going to f****** kill you.”
Instead of dealing with James, Mr Ebrahimi was arrested for breaching the peace. As he was led away, the crowd cheered and shouted “paedophile”.
He was released from custody the following day, July 12, and made 12 calls to police non-emergency number 101.
Mr Ebrahimi was informed that Duffy, his local beat manager, would visit but the officer refused to speak to him.
“My life is in danger. Right now a few of my neighbours are outside and shouting and calling me a paedophile. I need to see Pc Duffy,” Mr Ebrahimi told one operator.
Duffy told a supervisor: “He should be told in no uncertain terms that I will speak to him at my convenience. It’s Mr Bijan Ebrahimi. He’s well known to me and I won’t be taking any calls from him.”
He asked Passmore to conduct a “bit of a foot patrol” around Capgrave Crescent at about 8.40pm.
Passmore later told murder detectives that he had spent 40 minutes on foot patrol on the estate and a further 20 minutes on the adjoining streets.
The jury convicted him of lying about this, accepting the prosecution’s allegation that he simply drove up and down in his police car for just two to three minutes.
On July 13, Mr Ebrahimi tried to contact Duffy and a colleague, Pc Leanne Winter. He phoned police at 00.14am on July 14 – about an hour before his murder.
Witnesses saw James repeatedly stamp on Mr Ebrahimi’s head before setting him alight at 1.35am with neighbour Stephen Norley.
A post-mortem examination found Mr Ebrahimi, who suffered from depression, died before he was set alight.
James was jailed for life for the murder, while Norley was sentenced to four years in prison for assisting an offender.
Duffy and Passmore were two of 18 officers and staff facing misconduct proceedings within the force.
Pcs Leanne Winter, 38, and Helen Harris, 40, were acquitted of misconduct in a public office during the same trial as Duffy and Passmore.
The IPCC is expected to publish its investigation findings into Mr Ebrahimi’s death at the conclusion of all disciplinary proceedings.
The judge said: “I cannot go behind the jury’s verdicts and it is with a heavy heart that in each of your cases I take the view that only a custodial sentence is appropriate.
“It doesn’t seem to me a proper consequence of your wrongdoing that the sentences need be long.
“You have already suffered greatly. You have already lost your careers and in each of your cases there is genuine justification for mercy.
“You must not bear the responsibilities for the wider failings in the police which were beyond your control.”
Members of Duffy’s and Passmore’s families gasped and exclaimed “Jesus Christ” as the sentences were passed.
Mr Ebrahimi’s sisters were also sat in the public gallery of court room one for the hearing.
In a statement Avon and Somerset Police said: “Firstly, it is important to acknowledge that at the heart of this case is Bijan Ebrahimi’s family who have had to endure protracted legal and procedural processes. They have done this with the utmost dignity and composure.
“Our sole aim is to fully understand the circumstances around Mr Ebrahimi’s treatment by our police officers and staff in the days leading up to his murder.
“We are at the beginning of a series of misconduct hearings and meetings involving 15 members of staff and officers.
“It’s crucial that these disciplinary proceedings are allowed to progress to their final conclusion without prejudice now that the criminal case has come to an end.
“As a consequence, we’re unable to comment any further to avoid any prejudice to the disciplinary matters.
“The gross misconduct hearings involving police officers will be held in public at Police HQ in Portishead in line with national guidance, and further details will be published five working days before they are due to begin.”
The court heard Duffy received an email from Pc Henrietta Staveley-Brown after she visited Capgrave Crescent on July 12.
In the email, Pc Staveley-Brown described how she spoke to Charlene Husher, the girlfriend of James, following Mr Ebrahimi’s arrest the previous night.
Ms Husher insisted to the officer that Mr Ebrahimi was a paedophile and claimed that police had deleted indecent images of children from his camera.
She wrote to Duffy: “I told her to stop right there. We were getting into dangerous ground that could go into vigilante issues.”
A different neighbour informed her Mr Ebrahimi was “a paedophile and a rapist”, she told Duffy.
That afternoon, Mr Ebrahimi made 12 calls to police and repeatedly asked to speak to Duffy.
“Some of them were not urgent but there was a developing theme of Mr Ebrahimi saying that he didn’t feel safe and that a crowd was outside,” the judge said.
The judge said that Mr Ebrahimi had described people outside his home asking “where’s your handcuffs?”.
But Duffy refused to speak to Mr Ebrahimi, stating to call operators that he told “a porky pie” and “fabricated” events in his favour.
“The jury found a serious failing in your omission to visit, contact or take the calls of Mr Ebrahimi,” the judge said.
“I take account that you were never directed to visit Mr Ebrahimi.
“I accept that you had a number of dealings with Mr Ebrahimi over the years. He appears to have some faith in you.
“Mr Ebrahimi was on occasions not an easy man to deal with. He could make mountains out of molehills in relation to his disputes with the community.
“There’s no evidence that your misconduct was racially motivated. It arose because you regarded Mr Ebrahimi as a nuisance and someone who could not be wholly trusted.”
The judge said he had “much doubt” as to whether any action by Duffy could have prevented Mr Ebrahimi’s death.
Instead, Duffy asked Passmore to carry out a “bit of a foot patrol” of the area.
Passmore drove up to the gates and looked through to the communal green of Capgrave Crescent before reporting that all was calm, the judge said.
“There was, in reality, no need to lie about the nature of your patrol,” he told Passmore.
“You did such because you felt the nature of the patrol might be criticised. You chose to tell deliberate lies in your statement to the murder inquiry team.
“I do not accept that you made an innocent mistake.”
IPCC Commissioner, Jan Williams, said both officers “w ere experienced in their roles, and should have served Mr Ebrahimi far better in how they responded at a time when he was highly vulnerable”.
Ian Stern QC, representing Duffy, told the court his client’s action had “no consequences” in Mr Ebrahimi’s death.
He said: “You are dealing with someone who has given dedicated public service over many years.
“He has lost his good character, his employment and a significant financial sum for him and his family.
“He is a broken man and he will not work obviously in the areas in which he has hitherto worked and the public will be all the more poorer for that.”
Michael Borrelli QC, for Passmore, said his client was the carer for his elderly mother and stepfather and suffered from low IQ and memory recall.
He said: “Whatever he has done by writing that statement that he signed he didn’t act either in a deliberate or reckless disregard to a vulnerable member of the community that he served.
“He is someone who has demonstrated that at his core he is a good man who has spent time serving others.”
Passmore served in the Army before joining Avon and Somerset Police as a PCSO, the court heard.