The Dutch winger, who joined from PSV this summer for £25million, struck twice for the Red Devils in their 3-1 Champions League play-off first leg win over Club Brugge at Old Trafford on Tuesday night.
Depay controlled the ball brilliantly to dance around his opponent before firing low and hard in the 13th minute, to equalise after Michael Carrick's earlier own-goal, before adding a second with a powerful curling finish on the stroke of half-time.
The Holland international then set up Marouane Fellaini for United's third with a pinpoint cross in injury-time.
After the match, Depay admitted he was satisfied with his performance but felt he should have had his first match ball for his new club after missing a gilt-edged chance to complete his hat-trick.
"I ended the last season with a lot of goals," he told BT Sport.
"If you come to a big club you have to score and do the job, and of course it's a nice feeling to score at Old Trafford, my first two goals. [But] tonight I will think about that last chance and I am a little bit disappointed. I'll forget it tomorrow."