The Baggies had looked in danger of slipping back into the relegation dog fight when other results went against them earlier in the day.
But Tony Pulis' side are seven points above the bottom three with three matches to go thanks to their second successive win at Old Trafford
The last time the club won back to back victories at United was in 1959.
Chris Brunt's second-half free-kick hit Jonas Olsson and flew past David de Gea to secure a win that was accompanied by an impressive defensive display from the Midlanders.
Robin van Persie, making his first start since February 21, missed a penalty shortly after Albion took the lead.
United were poor all afternoon
Van Gaal's fears about being unable to break down defensively-minded opposition came true.
Despite having Van Persie, Wayne Rooney, and Juan Mata on the pitch, United could not breach a disciplined Albion team, who played with Saido Berahino as a lone frontman throughout.
The result means United, who wore black armbands as a sign of support for Rio Ferdinand after his wife died of cancer, have lost three Premier League games in a row for the first time since 2001.
A few weeks ago, the Red Devils seemed certain to qualify for Europe's top club competition, but there are now a few doubts they will come up short.
Liverpool, who beat QPR on Saturday afternoon, are now just four points behind Van Gaal's men, who have Crystal Palace, Arsenal and Hull to play before the end of the season.
The afternoon started on a poignant note for those on the pitch who played alongside Ferdinand and the people who wanted to show their support for him in the stands.
Chants of "Rio, Rio" came from the terraces as the players took to the pitch and in the fifth minute, almost everyone inside the stadium rose to their feet and applauded the former United defender, who wore five for the majority of his 12-year spell with the club.
Darren Fletcher, had an early chance to mark his return to Old Trafford with a goal, but the former United midfielder blasted his volley over the bar and into the Stretford End.
That was one of only a few occasions West Brom crossed the half way line in numbers in the first half.
They spent most of the time camped in their own half, a nine-man defensive block organised into two rigid lines, with Berahino stationed far up the pitch on his own.
United had 79 per cent possession in the opening 10 minutes but they did nothing with it.
Groans were heard as early as the 12th minute as Chris Smalling passed the ball sideways rather than forward while on the half way line.
Ashley Young looked the most likely player to prise open the Baggies' back line
He tricked his way past his marker and fired towards goal but Van Persie got in the way and accidentally diverted the ball wide.
West Brom's only form attack in the first period was through the air
Joleon Lescott wasted a good chance to score a shock opener when he planted his header straight into De Gea's arms
Youssouf Mulumbu had another effort saved.
Van Persie was looking sharp on his first start for 10 weeks
The Dutchman rifled a low shot just wide from Young's pull-back and Gareth McAuley put in a last-gasp challenge to dispossess the former Arsenal striker.
Ander Herrera kept chipping balls into the box from 30 yards out but West Brom spat them back out
Even Marouane Fellaini could not get the better of Jonas Olsson and Craig Dawson.
Frustration got the better of the Belgian
He somehow avoided punishment for pushing Claudio Yacob in the face following a tussle for the ball.
Fellaini moved up front after the break to try to add some muscle to United's front line, but the move did not pay off as the hosts continued to toil.
They had plenty of possession but lacked penetration.
Then came the killer blow
Paddy McNair bundled over Fletcher and handed the Baggies a free-kick 20 yards out.
Brunt let fly with a powerful shot which flicked Olsson's left leg and flew into the net with De Gea stranded.
Van Gaal responded by bringing on Angel di Maria for Daley Blind.
United felt they should have had a penalty when Mata's cross struck Lescott's right arm, but Anthony Taylor waved play on.
The referee did give the hosts a spot kick soon after when Antonio Valencia's cross struck Berahino's hand
The Albion players insisted the striker was outside the area, but after talking to his assistant Peter Kirkup, he awarded the penalty.
Van Persie took the ball but Boaz Myhill dived to his left and palmed his shot wide
Myhill would just not be beaten and t he Albion stopper sealed the three points for the visitors at the death with two fine saves from Van Persie and Di Maria
Source : PA
Source: PA