Anthony Martial's earlier goal, his fifth from those same four games, had given them cause for encouragement towards the end of the first half, during which they fell behind with Callum Wilson's 11th-minute finish and were consistently put under pressure.
Callum Wilson's opening goal was an example of the lax defending and the former Coventry man was one of three Cherries players who could have prodded home unmarked after United's defence decided not to bother with marking.
Mourinho was infuriated with a sloppy first half, saying: "It was the whole first half".
"I think we were much better in the second half". We passed the ball well and there was really good intensity to our play. I feel like people watching this game will not believe how hard we trained this week.
Then in the second half it was the exact opposite. We improve some of the players that were nearly there but not there - Lingard, Alexis - these kind of players, that were already available but not yet at a good level. "It was much better". I love the song 'Attack! It's a small thing but we need to get it right.
Scientists reveal mysterious 'Oumuamua' object could be an alien spacecraft
The team termed the possible propulsion system a "lightsail of artificial origin". It was moving at 59,030mph when it was first tracked by scientists.
"We were ready for everything, I can't understand the mentality that brings me more white hair!"
Matuidi was also substituted during the Serie A game on Saturday after suffering a hip problem and the French worldwide is also a doubt to face United having undergone treatment on the injury yesterday.
It's the third time in four games United have conceded first but still got something out of the game.
The visitors missed a string of chances after the break but just when it seemed Bournemouth would hang on, Rashford popped up in the six-yard box to chest down Paul Pogba's cross and tuck the ball past goalkeeper Asmir Begovic.
"In the second half I knew by having the two brothers on the bench - Rashford and Lingard - I knew the brothers could come in and make an impact", the Portuguese manager said after the game, as reported by Manchester Evening News. The lead can be overturned against teams like Newcastle and Bournemouth, but big games are coming thick and fast and United need to be ready for them from the first minute.