Amad Diallo’s stunning header five minutes into the second half looked like giving United a first-leg lead.
But Milan, who had already been harshly denied a goal when Franck Kessie’s first-half volley was ruled out for handball, replied when Simon Kjaer evaded the attentions of Nemanja Matic at a corner right at the death and steered a near-post header past Dean Henderson’s weak attempt to save.
It was the least a Milan side lacking former United striker Zlatan Ibrahimovic deserved.
It was tough luck on Diallo, who scored in quite brilliant fashion with his first effort on goal in only his third appearance for the Red Devils.
The Ivorian raced on to Bruno Fernandes’ brilliant chipped pass five minutes after replacing Anthony Martial at half-time, then, with his back to goal close to the penalty spot, looped the ball over Gianluigi Donnarumma and into the net.