The 18-year-old Diallo, who joined the club from Atalanta in January, had made headlines with his displays with the United Under-23 side and made an immediate impression after coming on as a second-half substitute on Thursday, rising to guide a well-placed header out of the reach of Gianluigi Donnarumma in the 50th minute.
The tally was the Ivorian's first attempt at goal for United and made him the youngest non-British player to ever score in major European competition for the club.
United, who saw an 11th-minute Franck Kessie goal ruled out for handball by a VAR review, looked set to survive the match without allowing a critical away goal, but Kjaer lost his marker and powered his stoppage-time header hard and past Dean Henderson to give Milan a slight advantage heading into the second leg.
It was the first goal United had conceded in 503 minutes in all competitions and means they will have their work cut out in next week's second leg.
Speaking to BT Sport after the match, Manchester United boss Ole Gunnar Solskjaer said Kjaer's goal will make the return leg more difficult.
"That makes it harder but it was always going to be down to the second game," Solskjaer said. "Henderson can save it of course, but we should have attacked the ball.
"We were a long way off it today. We were too slow with the possession, it is sometimes hard off the back of a game like Manchester City."
On Diallo's performance, Solskjaer added: "Diallo is creative, plays with freedom. Amad a nice goal, he still has loads to learn and will be better for that."
United and Milan have clashed 10 times down the years in Europe, winning five apiece, but it was the first time they had locked horns since United claimed a 7-2 aggregate victory in the Champions League last 16 in 2010.
The return leg will take place next Thursday at the San Siro.