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.
Manchester United remain second in their Champions League group behind Juventus, but are desperate to get something from their visit to Italy to make qualification for the last 16 less tricky. "A tough time, I thought I was the luckiest manager in the Premier League.
We are entering a period where we have some really hard games but I would say every game is so tough at this level", added the Cherries boss. Absolutely very bad. And not just my defenders, but the team. With 20 points from 10 games, they might even expect to be in the Champions League places at this point in the season. We assume we'll score two or three goals every game but we can't always do that.
United remain outside of the Premier League's top four, but again have a sense of momentum and objective that had eluded them earlier this season, and 55-year-old Mourinho told BT Sport: "We were a very, very lucky team".
Bournemouth went close to taking the lead at the beginning of the second half, but David Brooks' impudent flick from close range was well saved by De Gea, who managed to grasp the ball at the second attempt.
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"The discussion at half-time was of course that we weren't winning but deserved to be winning by more than one".
"I think by the end of December, which is the end of the first half of the Premier League, I think we are not going to be in the position we are now".
"The approach [last season] was much more defensive and with a much more defensive approach you can hide better your weaknesses", Mourinho said.
"We can take the second half to Turin and I think we will be more on the front foot and you will see a better performance in the first half".
"Manchester United changed things and made substitutions, but I was pleased with my players attitudes to the game and they gave everything to the match".