Bidding to reach the group phase after a first season out of the competition in 19 years, United fell behind to an early Michael Carrick own goal, only for Depay to turn the game around before half-time with a pair of brilliant strikes.
Brandon Mechele was sent off for the visitors in the second half, but it was not until stoppage time, when substitute Marouane Fellaini converted a cross from Depay, that Louis van Gaal's side truly took control of the tie, which concludes in Bruges next week.
But it was the hordes in blue and black in the East Stand who were the first to leap from their seats as Club Brugge took a shock eighth-minute lead.
Victor Vazquez's free-kick from deep on the left flicked off Adnan Januzaj and in attempting to clear the ball, Carrick succeeded only shinning it past his own goalkeeper, Sergio Romero, from 16 yards.
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It was the first goal United had conceded this season, but they were level within five minutes as Depay opened his account for the club in style.
Old Trafford erupted and although Romero was required to field a volley from Dion Cools, United would not be sated.
Wayne Rooney had a volley hooked off the line by Vazquez and Depay squandered a one-on-one by trying to find Januzaj, only to overcook his pass, before warming Sebastien Bruzzese's palms from range.