United led the contest three times and produced their most fluent attacking display under Amorim, but defensive lapses and missed chances prevented them from turning dominance into victory. The result leaves United frustrated, even if the performance excited the home crowd.
Amad Diallo opened the scoring with a close-range header after Diogo Dalot's delivery caused problems, before Bournemouth responded through Antoine Semenyo, who fired in off the post just before the break. United regain took control on the stroke of half-time when Casemiro powered home a header from Bruno Fernandes' corner.
The second half swung wildly. Evanilson leveled inside a minute of the restart, before Marcus Tavernier curled in a superb free-kick to put Bournemouth ahead. Fernandes restored parity with a brilliant free-kick of his own, and Matheus Cunha then pounced on a loose clearance to score just his second league goal for the club. Bournemouth refused to fold, with substitute Eli Junior Kroupi striking late to make it 4-4.
Amorim focused firmly on what United failed to do rather than what they created.
"We are really disappointed. It was a crazy game," Amorim said. "It might look like we lost two points in the second half, but I think we lost them in the first half.
"We dominate and create so many chances. We have to go to half-time with a different result. In the end we deserve more."
United's attacking play drew praise inside Old Trafford, but Amorim insisted entertainment must come with efficiency.
"People want to win, but they also want to be inspired," he said. "We play good football in many moments, but we need to be more clinical because we create so many chances."
Defensively, United missed the experience of Harry Maguire and Matthijs de Ligt, and they relied on goalkeeper Senne Lammens later on. The young keeper preserved a remarkable club record by making two stoppage-time saves from David Brooks, ensuring United still have not lost a home league match they are leading at half-time since 1984.
"Lammens does really well," Amorim added. "There are opportunities for both sides, but we create more. We need to find ways to close games. This has happened many times, and it is part of the process."
The point lifts United to sixth in the table, just two points off the top four, but the feeling at Old Trafford is clear. Under Amorim, United are playing with growing ambition and flair. Now, their head coach wants the cutting edge to match it.