Doctors at Papworth Hospital are very happy with how his new heart is performing, the union added.
Mr Serwotka contracted a virus in 2010, which caused his heart to fail. For the last two years he has been kept alive by a mechanical pump.
He was put on the urgent transplant list in August after developing a clot.
Mr Serwotka, who was raised in the south Wales valleys before becoming an Aberdare benefits officer, is now recovering in the Cambridgeshire hospital and will then recuperate at home.
Speaking to the BBC last month, Mr Serwotka, who is married with two children, described how he went from living a a very fit and active life to barely being able to walk 50 yards.
"We think it (the virus) was contracted whilst taking my dog for a walk in the woods. He came back smelling of something foul and so I washed him down."
"The next day my face and my legs had swollen grotesquely. It looked like an extreme allergic reaction.
"A week later I got rushed to hospital with a heart rate of 220 beats per minute.
"And after six weeks of investigation it was established I had contracted a virus that had caused my heart to massively inflate and swell, and when the swelling went down the heart had been irreparably damaged."
He was fitted with a heart pump, complete with battery pack and plug socket.
"I have an electric cable that comes from my heart, out through my abdomen, into a power pack, which is powered by batteries, but if I am in bed, I plug it into the mains. If I am driving the car, I plug it into the cigarette lighter," he told the BBC.
Mark Serwotka, general secretary of the Public and Commercial Services Union, has had a successful heart transplant, the union said.