Stevanovic, 17, will cost United an initial €12m and will remain at Partizan on loan for a year.
United bosses had a chance to see Stevanovic close up last November when he appeared for 35 minutes against them in the Europa League.
That was enough to convince them to invest in the young Serb, a left winger who can also play on the right.
Stevanovic has already made 42 first-team appearances for Partizan, scoring nine and assisting seven goals.
The teenager has pretty much confirmed the move on his Instagram account, with Old Trafford marked as his location.