Congratulating Ms Egan on her victory, Sir Keir said he looked forward to working with her.
The PM also praised Ms McAnea’s “outstanding contribution to the Labour and trade union movement”.
“She played an important role in securing the landmark Employment Rights Bill and her work has ensured this Labour government will bring in a fair pay agreement to boost the wages of social care workers,” he added.
With more than 1.3 million members working in sectors providing public services, including education, local government, the NHS and the police, Unison is the UK’s biggest trade union.
Ms Egan won just under 60% of the vote, although turnout was low, with fewer than 100,000 of the union’s members taking part.
She will begin her five-year term in January.
In a statement following her victory, Ms Egan said: “This result means ordinary Unison members are at long last taking charge of our union.
“We will put faith in members’ decisions and stand up to any employer, politician or cabinet minister who acts against our interests.”
Ms Egan’s election was welcomed by former Labour leader Jeremy Corbyn and his new left-wing group, Your Party.
A Your Party spokesperson said: “Andrea’s victory on an anti-austerity ticket shows that working-class people won’t take more cuts and more decline.
“Unison members have said no to Starmer’s crumbs off the table, no to chronic low pay, no to underfunding and no to privatisation.”
