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 Rashford, named in England’s provisional Euro 2016 squad a day earlier, and substitute Ashley Young secured the three points, but it was only enough to lift Louis van Gaal’s side above Southampton into fifth place.
Once again occupying a withdrawn midfield role, Rooney did his best to get United going by spraying passes around from deep, but it was not until the 32nd minute that the hosts threatened,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.
United went ahead two minutes before half-time with a goal completely out of keeping with what had come before.
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 with a shot from even further out.
Jesse Lingard also shot narrowly wide and Mata — omitted from Spain’s Euro 2016 squad earlier in the day — tested Federici after cutting inside onto his right foot and dispatching a low shot.
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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