The Dutch international pounced in the 13th and 43rd minute of the first-leg clash at Old Trafford to take the sting out of an eighth-minute away-goal for the Belgians courtesy of Carrick’s own-goal, before Maroune Fellaini netted a third in stoppage time.
Brugge's Brandon Mechele had been sent off in the 79th minute for a second bookable offence and United rallied late on to make the tie more comfortable, with Fellaini popping up right on cue.
The Red Devils now travel to Belgium on August 26 for the second leg with a two-goal cushion.
In another high-profile qualifier, Italian club Lazio beat German side Bayer Leverkusen 1-0 at home courtesy of a 77th-minute goal from Balde Diao Keita.
UEFA Champions League qualifying results
FC Astana 1-0 APOEL Nicosia
BATE Borisov 1-0 Partizan Beograd
Lazio 1-0 Bayer Leverkusen
Manchester United 3-1 Club Brugge
Sporting CP 2-1 CSKA Moscow
