Antonio Conte's side failed to restore the seven-point gap between themselves and second-placed Tottenham as goals from Marcus Rashford and Ander Herrera gave Jose Mourinho the bragging rights over his former employers.
It represented a rare setback in what has at times appeared to be a title procession for the Blues, but Begovic – who replaced the injured Thibaut Courtois for the clash at Old Trafford – is confident they will fend off their in-form London rivals.
"I think the defeat is down to a combination of things. United were more aggressive, they won second balls and probably wanted it more," Begovic told BBC Sport.
"They got a great start and then went on from there. We need to be better going forward.
"We have a lot of match-winners. We weren't at it, but we have a lot of character and can bounce back.
"We have a four-point advantage and at the start of the season we would have taken it at this stage.
"If we win our games it doesn't matter what Tottenham do."
Chelsea and Tottenham take a break from Premier League action next time out, as they meet at Wembley in the semi-finals of the FA Cup.