The Reds went a goal down midway through the first half in a frenetic couple of minutes - with a VAR intervention briefly threatening to reduce Bournemouth to 10 men.
But after a yellow card was deemed sufficient punishment for Tyler Adams' challenge on Alejandro Garnacho, Bournemouth immediately went on the attack and scored through Antoine Semenyo's low drive.
United were pretty dominant for much of the game thereafter, going closest in the first half when Garnacho forced Kepa Arrizabalaga to tip wide from a one-on-one opportunity. But for Dango Outarra striking the post soon after the half-time interval, United threatened plenty in the closing 45 minutes, with 25 shots in all.