A poor start to the first half saw them trail early on to an Amad goal, before some Antoine Semenyo brilliance drew Bournemouth level.
Casemiro scored in first half stoppage time to hand the hosts the lead before the break, but a flying start to the second half saw Bournemouth take the lead thanks to an Evanilson goal and a Marcus Tavernier free kick.
But United came back into it and turned the game around, Bruno Fernandes and Matheus Cunha scoring in the 77th and 79th minutes to give the hosts the 4-3 lead.
However, the Cherries weren’t done there – substitute Junior Kroupi found the back of the net with six minutes left to play to equalise at 4-4 and earn the Cherries a point.
Bournemouth's Eli Junior Kroupi celebrates scoring their side's fourth goal of the game during the Premier League match at Old Trafford, Manchester. _(Image: Martin Rickett/PA Wire)_
Andoni Iraola made one change to his side which drew 0-0 with Chelsea, as Tyler Adams returned from suspension in place of Alex Scott, who dropped to the bench.
The hosts had an early shout for a penalty waved away after they broke at pace and Bruno Fernandes shot, but it was deflected off Bafode Diakite and into Adrien Truffert, but his arm was in front of his body and nothing was given.
The Cherries were dealt an early blow as Adams was forced off after just five minutes after a collision with Casemiro as both went for a bouncing ball, and he was replaced by Scott.
United made the much brighter start and Djordje Petrovic was called into action to deny Mason Mount, after he powered towards goal from a Brian Mbeumo cross.
Manchester United's Amad Diallo scores their side's first goal of the game during the Premier League match at Old Trafford, Manchester. _(Image: Martin Rickett/PA Wire)_
And the Red Devils made their good start count as they took the lead with 13 minutes on the clock.
Matheus Cunha drifted between the two Cherries centre backs as a cross came from Diogo Dalot on the left and the ball dropped over his head.
Petrovic was unable to get the decisive touch, and Amad was on hand to head the ball over the line from less than a yard.
After a brief VAR check, the goal was given.
United continued to come forward, and Cunha put a good chance wide of the top corner when he found space on the edge of the box.
Mbeumo was the next to have a shot on goal, but Petrovic saved well to his left, before the Cherries went close to an equaliser.
Justin Kluivert delivered a good cross from the right and Marcus Tavernier arrived from deep to meet it – but he could only head at Senne Lammens and the ball bounced clear.
Adam Smith denied Amad a second goal as he slid in brilliantly to block his shot from the centre of the box following a quick counterattack.
Mbeumo went close with an acrobatic volley but he could only fire over the bar, before Casemiro shot wide.
Bournemouth's Antoine Semenyo celebrates scoring their side's first goal of the game during the Premier League match at Old Trafford, Manchester. _(Image: Martin Rickett/PA Wire)_
The Cherries drew themselves level through a brilliant Semenyo goal five minutes before the interval.
Luke Shaw was barged off the ball by Justin Kluivert and the ball dropped to Smith, who played it into the Ghanaian. He then drove towards the United box and fired into the bottom corner past Lammens.
Tavernier drew a good save out of the United keeper with a deflected shot from distance.
But United retook the lead four minutes into first half stoppage time. Fernandes delivered a corner to the back post, where Casemiro had managed to get free, and he headed past Petrovic – the Serbian should have done better.
Manchester United's Casemiro celebrates scoring their side's second goal of the game during the Premier League match at Old Trafford, Manchester. _(Image: Martin Rickett/PA Wire)_
It was the perfect start to the second half for the Cherries, though as they drew themselves level just one minute into the half.
Tavernier collected the ball from a Senesi header inside the United half and he turned into space, before playing a good ball through to Evanilson.
The Brazilian shot back across Lammens and into the bottom corner for his first goal since the trip to Tottenham in August.
And the Cherries took the lead just six minutes later.
Tavernier won a free kick as he drove towards the United box and the Englishman stepped up to curl the set piece into the back of the net from 25 yards.
Bournemouth's Marcus Tavernier scores their side's third goal of the game during the Premier League match at Old Trafford, Manchester. _(Image: Martin Rickett/PA Wire)_
United sought a way back into the game. Mbeumo blasted a shot over the bar with 20 minutes left to play, after Petrovic had tipped a sliced Cunha cross onto his crossbar and over.
Cunha stung the palms of Petrovic, with Smith giving away a free kick for handball just outside the box as the ball dropped.
From the resulting set piece, Fernandes picked out the top corner off the underside of the bar to draw United level.
Kroupi replaced Evanilson after the goal, before United retook the lead.
Manchester United's Bruno Fernandes scores their side's third goal of the game during the Premier League match at Old Trafford, Manchester. _(Image: Martin Rickett/PA Wire)_
The hosts came forward at speed, Benjamin Sesko took the ball off Mbeumo and played the ball across the top of the box.
Truffert was unable to deal with it and it fell to Cunha with just Petrovic to beat, and he did the rest to give the hosts the advantage with 11 minutes to play.
Substitute Kroupi put Bournemouth back on level terms five minutes later.
Fantastic work from Jimenez saw the young Spaniard drive towards the box and pass to the Frenchman, who finished past Lammens to equalise at 4-4 in the 84th minute.
Bournemouth's Eli Junior Kroupi celebrates scoring their side's fourth goal of the game during the Premier League match at Old Trafford, Manchester. _(Image: Martin Rickett/PA Wire)_
David Brooks and James Hill replaced Kluivert and Tavernier as the game entered eight added minutes.
Brooks drew a fantastic save out of Lammens with a diving header from a Jimenez cross deep into stoppage time, before Senesi shot straight at the United keeper.
Brooks again tested Lammens but couldn’t find the net as referee Simon Hooper called time on a dramatic game of football.
The point sees the Cherries remain 13th in the Premier League, on 21 points.
**Manchester United:** 31, S. Lammens; 23, L. Shaw (Dorgu 90+3’); 26, A. Heaven; 15, L. Yoro (Martinez 69’); 2, Diogo Dalot (booked, 43’); 8, Bruno Fernandes (c); 18, Casemiro (booked, 50’)(Mainoo 61’); 16, A. Diallo; 7, M. Mount (Sesko 69’); 19, B. Mbeumo (Zirkzee 90+3’); 10, Matheus Cunha.
**AFC Bournemouth:** 1, Đ. Petrović; 3, A. Truffert; 5, M. Senesi (booked, 76’); 18, B. Diakité; 15, A. Smith (c)(booked, 79’); 12, T. Adams (Scott 5’); 16, M. Tavernier (Brooks 90+1’); 24, A. Semenyo (booked, 43’); 19, J. Kluivert (Hill 90+1’); 20, Álex Jiménez; 9, Evanilson (Kroupi 78’).