Shortly after Michael Carrick had put Brugge ahead with an own goal, Depay skilfully juggled a pass from the England midfielder before driving home his first goal for United.
The £25million summer signing then doubled his tally with a peach of a curling shot just before half-time.
United looked set to take a slender advantage to Belgium for the second leg despite Brandon Mechele’s dismissal, but Fellaini stepped off the bench to nod home in the 94th minute.
The return of the Champions League anthem after a 16-month absence raised spirits inside Old Trafford on Tuesday, but the home fans were soon stunned into silence thanks to Carrick’s own-goal.
The midfielder, who had put a simple pass out of play in the opening minute, fell to the floor attempting to clear Fernando Vazquez’s free-kick but instead guided the ball past a diving Sergio Romero.
Five minutes later, Carrick was back in the good books. The midfielder lifted a ball into the box to find Depay, who juggled the ball away from his marker before unleashing a low drive past Sebastien Bruzzese.
Wayne Rooney had a shot cleared off the line by Vazquez and then glanced a header just wide, but it was not all United as they left open holes at the back, which Brugge exploited on the break.
The Belgians could not find a way past Romero, though, and United made it 2-1 just before the break.
Depay cut inside the left flank on to his right foot and whipped a wicked shot past Bruzzese before performing a jig in the corner.
Van Gaal brought Bastian Schweinsteiger on for Carrick at half-time.
Depay started where he left off after the break, curling a low shot towards goal, but this time Bruzzese saved.
The 21-year-old should have had his hat-trick just after the hour when Rooney’s backheel fell straight into his path, but he put the ball into the Stretford End.
Brugge still looked threatening on the counter. Romero dallied on the ball after a poor backpass from Matteo Darmian, but the referee stopped play because Mamadou Oulare had stood on the goalkeeper’s ankle as he attempted to capitalise.
Mechele, who had been cautioned earlier for a foul on Depay, put his team’s chances of a late equaliser in jeopardy when he slid in on substitute Javier Hernandez to receive his second booking.
Van Gaal brought Marouane Fellaini on for Rooney with six minutes to go and the move paid off as the big Belgian powered in Depay’s cross to make it 3-1.