Romelu Lukaku's ninth league goal of the season, a 25th- minute header at the back post from a Juan Mata cross, was enough to give second-placed United the win but they had keeper David De Gea to thank for the three points.
The Spanish keeper kept out efforts from Charlie Williams, Dan Gosling and Ryan Fraser and blocked a fierce strike from substitute Jermain Defoe eight minutes from time as Bournemouth pressed for an equaliser that would not have been undeserved.
The nearest United came to killing the game off was a thundering long-range effort from Marcus Rashford in the 72nd minute which crashed against the bar.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
Although United, with 21-year-old Scott McTominay in midfield, rarely found much fluency on a rainy and cold night their manager was happy with the win.
"I am pleased with the three points, it was difficult match. They had one more day’s rest than us, they were fresher than us, and mentally fresher because a big match takes more from our players," Mourinho said.
The United boss denied his team had a hangover from the derby loss to City on Sunday but said it had clearly taken a toll on his players.
"Fatigue, yes, but hangover - no. The game was a big one, and a defeat does not help in your recovery - victories help, defeats don’t. They were difficult opponents and difficult conditions. If we had scored the second it would have been a calm night for us," he said.
United remained 11 points behind leaders Manchester City who won 4-0 at Swansea City and Mourinho said his team would continue to try and catch up.
"I said we play match after match and the next we have to try to win, we have a difficult one at West Brom and we try match after match. It is only over in May, if it is over now I go on holiday to Brazil or Los Angeles," he said.
Bournemouth boss Eddie Howe felt his side deserved more from the game.
"I’m very disappointed, I thought we were excellent. We created opportunities, our defensive shape was very good and we deserved a lot more today," he said.
(Reporting by Simon Evans, editing by Ed Osmond)
