The defeat to City left United trailing Pep Guardiola's side by 11 points and ended United's 40-game unbeaten run at home and Jones believes that his team-mates showed immense character to bounce back at the first time of asking against the 'Cherries'.
"It was always going to be difficult off the back of the derby defeat," said the England man. "[We were] so disappointed in many ways because of the magnitude of the game and we wanted to do more.
"We wanted to win the game and unfortunately it wasn't to be, so it was important that we came out tonight, showed a bit of fight and a bit of courage, [with] players getting on the ball and trying to do something.
"It wasn't pretty at times. We had to dig deep and grind it out, but the most important thing was to get three points and get back on track.
"I'd just like to say to thank you to the fans as well because I think it was difficult for them on Sunday in the derby defeat.
"Tonight they stuck by us again. When times are hard, they're singing, they're chanting for us and we'll need that going on in the season."
Romelu Lukaku scored the only goal of the game against Eddie Howe's side, a triumph of sorts after being criticised for his performance against City and speaking to 'MUTV', Jones said that the goal was typical Lukaku and that he expects the Belgian to score plenty more over the festive period.
"He (Lukaku) did well, rose above his man and that's what he does best," Jones explained.
"He's going through a patch where he's not scoring goals, but he's doing other things for us in the match. I am pleased for him tonight. Hopefully he can kick on and score some more goals now."