Playmaker Juan Mata was the villain of the piece after his first-half dismissal forced United into playing more than an hour with 10 men, while their defence eventually creaked as in-form Salomon Rondon netted the winner.
Back-to-back victories had moved United level on points with fourth-placed Manchester City, although defeat, coupled with victories for Manuel Pellegrini’s outfit and West Ham saw the Reds slump to sixth, three points adrift.
United can qualify for next season’s Champions League by winning the Europa League this term – they face Liverpool in the first leg of their last-16 tie on Thursday -- although Van Gaal has not given up on the more conventional route with nine matches of the domestic campaign remaining.
“It’s a big blow, these kind of matches you have to win. We need 27 points,” said Van Gaal. “It is a lot, but it is always possible.
“You always have a chance [to make the top four] but these kind of matches you have to win, and certainly when you have to close the gap because you have lost too many matches already.”
Van Gaal also suggested that Mata’s previously sparse rap sheet should have excused him a red card.
The 27-year-old former Chelsea midfielder had never previously been sent off in the Premier League. But the Spaniard was booked twice in the space of three minutes, firstly for needlessly blocking a Darren Fletcher free-kick and then for a foul on his ex-United team-mate.
“The referee has to know the person who makes the tackle and Mata never hurts an opponent,” said Van Gaal. “You can give two bookings according to the rules but I think the referee needs to know the player who is doing that.”
Venezuela striker Rondon proved the matchwinner in the 66th minute as he controlled a cross from substitute Sebastien Pocognoli before lashing low past goalkeeper David de Gea as West Brom recorded their first home league win over United for 32 years.
Victory propelled the Baggies to 39 points and moved them above Watford and Everton to 11th, 15 points clear of struggling Norwich and the relegation zone.