Sir Keir, who replaced Jeremy Corbyn as leader on Saturday after convincingly winning the leadership contest, spoke to Jewish community leaders over Skype, along with newly-elected deputy leader Angela Rayner.
The former shadow Brexit secretary apologised for Labour's failure to deal with anti-Semitism in its ranks, calling it a "disgrace", and promised to establish new procedures to deal with fresh cases.
The issue dogged Labour after Mr Corbyn's election as leader in 2015, and leadership hopefuls were regularly asked how they would tackle it.
An inquiry into Labour's handling of the crisis is currently being undertaken by the Equality and Human Rights Commission (EHRC).
Sir Keir, in his online call on Tuesday, said he had demanded that all the outstanding anti-Semitism cases be "on my desk at the end of the week", while also vowing to set up an independent complaints process.
The move was praised by Leeds North East MP Fabian Hamilton, who said the process would “go some way to rebuilding that trust, as will a firm commitment to implementing any recommendations from the Equality and Human Rights Commission (EHRC), when their report comes”.
He said “actions always speak louder than words” and that after the General Election he “extended the hand of friendship to the Leeds Jewish Representative Council, in the spirit of reconciliation and togetherness”.
Mr Hamilton said: “Keir is right when he says ‘if you are anti-Semitic, you cannot and should not be in the Labour Party. No ifs, no buts’. We will fight to rid our party of this evil scourge, wherever it rears its ugly head.
“In time, I hope those who have made the difficult decision to leave our party because of anti-Semitism feel they are now able to come back.”
The Board of Deputies president Marie van der Zyl, Jewish Leadership Council (JLC) chairman Jonathan Goldstein, Community Security Trust (CST) chairman Gerald Ronson and Jewish Labour Movement (JLM) chairman Mike Katz issued a joint statement following the online meeting with the new Labour leader.
They said: "While we would have fully understood the need to focus entirely on coronavirus at this time, Keir Starmer has already achieved in four days more than his predecessor in four years in addressing anti-Semitism within the Labour Party.
"As we discussed with Keir and Angela, we want to have a normal relationship with Labour whereby we can discuss the full range of issues affecting our community, from religious freedom to Israel, from Jewish schools to poverty, from refugees to the environment - and not just anti-Semitism.
"This has certainly been a good start."
Sir Keir thanked the representatives for meeting with him "not least so close to Passover".
"It was very important to me to seek to address the disgrace of anti-Semitism in our party as soon as possible," he said.
"Today, I repeated once again the apology I made as soon as I was elected leader.
"Over the last few years, we have failed the Jewish community on anti-Semitism.
"Labour is a proudly anti-racist party and, going forward, it will not be enough to 'pass the test' on anti-Semitism. We need to set new standards for best practice.
"At today's meeting, I committed to begin work on setting up an independent complaints process, co-operating fully with the EHRC's inquiry and asking for a report on all outstanding cases to be on my desk at (the end of) the week.
"I also discussed with the Jewish Labour Movement my ambition to roll out training of all Labour Party staff as soon as practically possible."