Martial though hauled United level before half-time with an excellent run and finish after pouncing on a defensive lapse from Brandon Mechele.
United boss Ole Gunnar Solskjaer made six changes to the side that beat Chelsea on Monday, with Romero handed another start in Europe while Victor Lindelof returned from illness to form a three-man defence alongside Harry Maguire and Luke Shaw. New signing Odion Ighalo again began on the bench.
De Cuyper nearly popped up with a goal inside three minutes when Dennis escaped Maguire and skipped towards Romero, his attempt to pull back resulting in Nemanja Matic hooking against the teenager and the ball ricocheting fortuitously over the bar.
A long kick forward caught United short in defence and Dennis, who scored twice against Real Madrid at the Bernabeu in October, outmuscled Brandon Williams before lofting over Romero with the Argentine well off his line.
An increasingly ragged United were fortunate not to fall further behind when Mats Rits scuffed wide before Romero positioned himself expertly to keep out Dennis from a tough angle.
Martial takes advantage
Brugge then paid a heavy price when Mechele allowed a bouncing ball to run beyond him, an alert Martial seizing on the opening to race clear and steer calmly beyond Mignolet.
Jesse Lingard stabbed wide while lunging to reach a teasing ball from Diogo Dalot before Dennis was again denied by Romero from a tricky position.
Odilon Kossounou squandered a superb chance to put Brugge back on top when he scuffed wide following terrific work on the right from Dennis and a clever dummy from Siebe Schrijvers.
Bruno Fernandes injected fresh life into United as he was introduced for the final 10 minutes, and although the 2017 champions were unable to land a knockout blow they will fancy their chances of finishing off the tie on home soil.
Celtic keeper Fraser Forster saved a late Jens Stage penalty as the Scottish champions drew 1-1 away to FC Copenhagen after Odsonne Edouard’s opener was cancelled by an effort from Senegalese striker Dame N’Doye.
La Liga surprise package Getafe defeated last season’s Champions League semi-finalists Ajax 2-0 in Spain with Brazilian duo Deyverson and Kenedy on target.
Record five-time winners Sevilla needed the 82nd-minute equaliser from January arrival Youssef En-Nesyri to salvage a 1-1 draw away to Romanians Cluj.