Hull were not the only side to suffer a relegation setback with Middlesbrough joining them in the bottom three after being sunk by a Marko Arnautovic double in a 2-0 win for Stoke city.
There were dramatic victories for fellow strugglers Crystal Palace, 2-0 at West Bromwich Albion, and Paul Clement's resurgent Swansea City, who came from behind to beat Burnley 3-2.
After an incident-packed affair at Old Trafford, United manager Jose Mourinho was left with no complaints after his side again failed to take maximum points from a game they dominated.
After a Marcos Rojo goal was answered by a Josh King penalty in a turbulent first half, United were left sixth on 49 points, having missed the chance to overhaul both Liverpool and Arsenal, who play each other later on Saturday.
Ibrahimovic, fresh from scoring twice in United's Wembley triumph over Southampton, had a wretched afternoon and not just because his second-half penalty with the score at 1-1 was saved at full stretch by Artur Boruc, who gave an inspired performance.
Just before halftime, the Swede was fortunate not to be dismissed for elbowing Nathan Mings, moments after the Bournemouth defender stood on his head as he lay on the pitch.
In the subsequent fracas, Bournemouth's Andrew Surman pushed Ibrahimovic over and was sent off for this second yellow card offence. But United failed to convert their numerical advantage and dominance into goals.
Ibrahimovic denied he had acted out of retaliation. "My intention was not to injure him but to protect myself," he said.
Mourinho said United were to blame for missing out on three points. "Who can I blame? Ourselves. Nobody else. We missed big chances. We felt pressure of the clock running down and lost the quality of our attacking football," he said.
At the other end of the table, Hull remain on 21 points, one adrift of Middlesbrough, while Palace jump to 17th on 25 points after dazzling goals from Wilfried Zaha and Andros Townsend earned Sam Allardyce back-to-back wins.
Swansea's revival under Clement gained more momentum as Spanish international Fernando Lorente headed home twice, including a 92nd-minute winner, to earn them a fourth win in six games and move them on to 27 points, alongside Leicester and Bournemouth.
In the day's best match, Nathan Redmond struck twice and Manolo Gabbiadini scored a sixth goal in four games as Southampton edged Watford 4-3 in their Premier League clash at Vicarage Road.
(Reporting by Ian Chadband, editing by Neil Robinson)
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