Starmer  also sought to draw attention to his Labour government's record and away from "all the other discussions that are going on" in apparent reference to the speculation that has dogged his leadership since his party's mauling in England's local elections.
Asked whether he would be out supporting Greater Manchester mayor Burnham's campaign for the June 18 contest, Starmer told reporters on a visit to Essex: "Yes, and I've said to the whole Labour movement that I want everybody to be involved in the campaign, whatever other discussions are going on, it's really important - that's a straight fight between Labour and Reform.
"It's about the values that we take into politics, and it's really important that we all pull together and fight in that by-election, as we will do."
Starmer also talked up recent good news when asked whether he was frustrated by the focus on Burnham and former health secretary Wes Streeting's ambitions to oust him.
Calls from Labour MPs for him to go and a slew of resignations, mostly from the junior government ranks, last week threw Starmer's political future into doubt.
"There's been a lot going on in the last two weeks but what is important for me to draw out is that actually the last week, in terms of what we've delivered, is really significant," he said.
He pointed to recent figures showing the United Kingdom's real GDP grew by 0.6 per cent in the first quarter from January to March, falling National Health Service waiting lists and a drop in UK net migration to the lowest level since the start of the coronavirus pandemic.
The prime minister sought to regain control of the political agenda with a series of announcements on the cost of living on Thursday.
Chancellor Rachel Reeves set out a cut in the rate of tax on tickets for theme parks, zoos and museums from 20 per cent to five per cent over the summer holidays as part of the package to ease the effects of the Iran war.
Other measures included free bus travel for children during August, a 10 pence per mile increase in tax-free mileage rates and the cutting of import tariffs on more than 100 types of food products.
In response to Starmer's pledge to campaign for Burnham in the Greater Manchester constituency, a spokesperson for the mayor said: "Anyone who wants to embrace Andy's campaign message is welcome on the campaign."
Allies of Burnham have suggested he may not launch a bid for the top job immediately if he is successful in his attempt to return to parliament.
But a contest is widely expected after the mayor's call for a "new path" and promise to "change Labour".
Starmer has previously insisted he will not "walk away" from Downing Street.
Streeting, who quit as health secretary last week calling on Starmer to go, earlier rolled the pitch for a potential bid by setting out plans for a wealth tax in which capital gains tax would be equalised with income tax.
Luke Charters became the latest Labour MP to quit as a government aide on Thursday.
The MP for York Outer said Labour must be "bolder" and "more confident" as he announced he had resigned as a parliamentary private secretary for Business Secretary Peter Kyle in a post on X.
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