Koeman has worked with Stekelenburg on two separate occasions while he was in charge of Ajax Amsterdam and last season when the 33-year-old was on loan at Southampton.
"I'm very happy - things have moved very fast over the last couple days," Stekelenburg told EvertonTV.
"He [Koeman] has come to Everton and he called me a few days after asking if I was willing to follow him. It wasn't difficult to make that choice.
"He's one of the best coaches - he's straight talking, he's fair and he has confidence in me which counts as well.
"I've been here [in England] for a couple of seasons and I love the Premier League. It's fantastic that I can stay."
Koeman was equally pleased to be reunited with the goalkeeper, whom he holds in very high regard.
"Obviously I know Maarten well having worked with him at two clubs before," added the Toffees manager.
"I know through his professionalism, hardworking approach and personality he will fit in well with what I will be building at Everton."