Two goals in the space of eight first-half minutes from Romelu Lukaku and Jesse Lingard sent the visitors into the break in complete control, but they were forced to hang on at the end of the contest after Gareth Barry had pulled a goal back.
The hosts ultimately fell short of ending what is now a club-record 17-match winless streak across all competitions, with United holding on for a third consecutive away win and a sixth victory in their last seven outings.
United had seen the gap at the top of the table grow to 14 points courtesy of City's victory over Tottenham Hotspur on Saturday, while Chelsea's win over Southampton also saw them draw level on points with Jose Mourinho's side.
Nevertheless, there was a lack of urgency in the visitors' play during the opening exchanges despite coming up against a West Brom team who had slipped into the relegation zone following Saturday's results.
Indeed, it wasn't until the opening goal shortly before the half-hour mark that either side had a meaningful chance, and United took full advantage of it as Lukaku - who scored a hat-trick for West Brom in his last appearance in this fixture - glanced a fine header past Ben Foster from Marcus Rashford's cross.
The hosts responded well to falling behind and it took an important interception by first Chris Smalling and then Antonio Valencia to deny Salomon Rondon and James McClean shortly afterwards, but United soon took the game beyond the Baggies.
Mourinho's side needed a big slice of luck to double their advantage, though, as Lingard's tame effort from the edge of the box was on course to be comfortably saved by Foster before it took a big deflection off Ahmed Hegazi and ended up in the back of the net.
It took until the 44th minute for David de Gea to be called into action for the first time, and even then it was a routine stop for the United goalkeeper as he got down to keep out Jake Livermore's powerful strike.
However, the Baggies did pose more of an attacking threat in the opening stages of the second half following Barry's introduction at the break and Alan Pardew's subsequent switch to a 4-4-2 formation.
Kieran Gibbs dragged one effort wide and Barry sent a tame shot straight at De Gea before Oliver Burke's hooked delivery into the middle was brilliantly cleared by Valencia under pressure from Rondon and McClean.
United soon responded, though, and Rashford drilled one strike wide of the far post before Lingard fired straight at Foster at the end of a quick counter-attack.
The visitors' control of the game began to loosen as the second half wore on, though, and West Brom fired a few warning shots before pulling a goal back as Rondon and Chris Brunt both squandered decent sights of goal.
United's increasing lethargy was eventually punished by an unlikely source as Barry scored his first goal for the club, passing his finish into the bottom corner from close range after Mourinho's side had failed to deal with an inswinging corner.
It was also the first goal of Pardew's reign to end a three-match streak without finding the back of the net, but West Brom wasted little time in pushing forward for another and completing a memorable comeback.
Jay Rodriguez looked most likely to provide it with a header which left De Gea rooted to the spot only to flash narrowly past the post, although United did come close to killing the game off in stoppage time when Lukaku dragged an effort wide of the target on the break.
In the end the visitors held off the late pressure from the Baggies, though, sealing all three points as West Brom's wait for a Premier League win stretched to 16 games.
West Brom (4-3-3): Foster; Nyom, Hegazi, Evans, Gibbs; Livermore, Yacob (Barry, 46'), Krychowiak (Brunt, 72'); Burke (Rodriguez, 64'), Rondon, McClean
Man Utd (4-2-3-1): De Gea; Valencia (Rojo, 66'), Smalling, Jones, Young; Herrera, Matic, Mata, Lingard (McTominay, 86'), Rashford (Martial, 79'); Lukaku