Amad opened the scoring and, after Antoine Semenyo levelled, Casemiro’s header ensured the Reds went into the break with a deserved lead.
Two goals in the opening seven minutes of the second half from Evanilson and Marcus Tavernier saw the pendulum swing in favour of the Cherries.
But Ruben Amorim’s side roared back with two goals in as many minutes, courtesy of a stunning Bruno Fernandes free-kick and a second United goal for Matheus Cunha.
The visitors, however, grabbed an 84th-minute equaliser through substitute Junior Kroupi to ensure the spoils were shared at Old Trafford on Monday night, much to United’s frustration.