Sarwar also warned fellow Unionists against mocking or crowing if the Supreme Court rules an IndyRef2 bill illegal.
The Glasgow MSP's comments were made as Labour opened up a massive poll lead against Liz Truss's Tories.
Relations between the UK and Scottish Governments are poor, with sniping made by both sides on a regular basis.
The Truss Government's outright refusal to grant a second referendum is another source of tension.
Although Labour is opposed to IndyRef2, Sarwar pointed to a more constructive relationship if Starmer becomes prime minister, saying: "We do want to have a relationship that's based on cooperation, not conflict.
"And at the moment, devolution isn't working because you have bad actors on both sides.
"You have a UK Government that actually probably doesn't believe in devolution.
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