Louis van Gaal was looking to Wayne Rooney to provide the goals at Old Trafford, but instead it was Fellaini and Depay who stepped up to the plate in this 3-1 first-leg win.
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
The forward barely celebrated
He knew there was much more work to do.
Rooney led the hunt for a second
The England captain had a shot cleared off the line by Vazquez and then glanced a header just wide.
It was not all United
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.
TWEET OF THE MATCH
Fitting in at @ManUtd like a Frenchman with his collars up! Well played @Memphis #ChampionsLeague #becurious - Lord Of The Rings actor Dominic Monaghan
PLAYER RATINGS
Manchester United:
ROMERO: 6
DARMIAN: 6
SMALLING: 7
BLIND: 6
SHAW: 7
CARRICK: 6
SCHNEIDERLIN: 6
MATA: 6
JANUZAJ: 6
DEPAY: 9
ROONEY: 5
Substitutes:
SCHWEINSTEIGER: 6
HERNANDEZ: 5
FELLAINI: 7
STAR PLAYER
Memphis Depay: Looked a bit wooden at times during his first two games, but showed why he is regarded as one of Europe's most promising talents on Tuesday night
The Dutchman cut through Brugges' defence with his speed from the first whistle
He showed sublime skill to juggle the ball in the build up to his first goal and his second was a peach of a curler.
MOMENT OF THE MATCH
Memphis Depay's breathtaking second goal, which brought almost everyone inside Old Trafford to their feet
The Dutchman curled a perfectly-placed curling shot beyond the Brugge goalkeeper just before half-time.
VIEW FROM THE BENCH
Depay played in the number 10 role on tour and in the first match of the season against Tottenham, but he has looked far better since moving out to the left flank
The 21-year-old reminded Louis van Gaal how devastating he can ben when cutting in off the wing on Tuesday night
Van Gaal's opposite number Michel Preud'homme did the best he could considering he had eight players out injured.
MOAN OF THE MATCH
Brandon Mechele's sending off
The defender had played well all match before fouling Depay to enter the book in the 73rd minute
Then, just six minutes later, he clattered into Javier Hernandez right in the corner and referee had no choice but to issue a second caution.
WHO'S UP NEXT
Manchester United v Newcastle (Premier League, Saturday 22 August)
Waregem v Club Brugge (Belgian Jupiler League, Saturday 22 August)
Source : PA
Source: PA