In a game carried over from Sunday following a postponement caused by the discovery of a fake bomb accidentally left behind during a terror training exercise, United needed a 19-goal victory in order to steal past Manchester City into the fourth and final Champions League qualifying berth.
Goals from Wayne Rooney, 18-year-old Marcus Rashford, named in England's provisional Euro 2016 squad a day earlier, and substitute Ashley Young secured the three points yesterday, but it was only enough to lift Van Gaal's side above Southampton into fifth place.
As Bournemouth's fans goaded him with chants of "You're getting sacked in the morning!", Van Gaal added: "You never let us down. Thank you for that. We will bring the FA cup home to you. You deserve it. Thank you."
The result secured a Europa League group place for United, while Saturday's FA Cup final against Crystal Palace may yet yield a piece of silverware.
But it was a tame end to the season, at a two-thirds full stadium, which laid bare United's stagnation under Van Gaal. The lap of appreciation that followed his address was cursory.
The south-coast club, who replied courtesy of a stoppage-time Chris Smalling own goal, finished their first ever top-flight campaign in 16th place.
But Howe said that Sunday's events had proved problematic.
"It was quite difficult, mentally," he said. "We did our best, but Man United were too good for us."
Fans arriving at Old Trafford spoke of their anger and frustration over Sunday's cancellation, which left visiting supporters facing two 500-mile (800-kilometre) round trips in three days.
Once again occupying a withdrawn midfield role, Rooney did his best to get United going by spraying passes around from deep, but it was not until the 32nd minute that the hosts threatened, with Michael Carrick just failing to get a telling touch on Rooney's free-kick.
Anthony Martial, as so often, was the instigator, swapping passes with Juan Mata and cutting the ball square towards Rashford, whose clever dummy allowed Rooney to tuck home his first goal since February 2 and his 100th in the league at Old Trafford.
United emerged from their slumber after the interval.
Mata was unfortunate not to win a penalty after Steve Cook bundled into him from behind, while Carrick rattled the bar with a deflected 25-yard effort and Antonio Valencia drew a soaring save from Bournemouth goalkeeper Adam Federici from even further out.
With 16 minutes to play the hosts made it 2-0. Rooney's flighted pass was nodded back by Valencia and Rashford beat Federici with a crisp, low strike for the eighth goal of his 17-game United career.
Rooney released Young to smuggle a shot past Federici with three minutes to play, but there was still time for Smalling to shin the ball into his own net from Max Gradel's shot, depriving United goalkeeper David de Gea of a 16th clean sheet that would have seen him share the Golden Glove with Arsenal's Petr Cech.
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