Moments after the visitors had a penalty appeal turned down, they took a shock lead through Dominic Solanke's clinical close-range finish when there were only five minutes on the clock.
United dominated the first-half possession thereafter but Ten Hag felt some significant damage had already been done, even if Bournemouth still only led by one goal at half-time after Solanke hit the post and was also thwarted by a superb Harry Maguire challenge on the edge of the penalty area.
The Reds improved after the break until, against the run of play, the Cherries doubled their advantage through Phillip Billing's header and then went on to make it 3-0 through another aerial attack by Marcos Senesi.