England forward Rashford was given a straight red card 71 minutes in for pushing his head into Phil Bardsley, whose kick out at the 20-year-old had gone unseen by referee Jon Moss.
The dismissal did not prove costly to the visitors at Turf Moor, though, as United saw out the victory given to them by Romelu Lukaku's first-half double.
Mourinho felt Rashford was to blame for being provoked by former United man Bardsley, but he was nonetheless thrilled with his side's display.
"I don't know what happened," he told BBC Sport. "I would say it was naive. It was a kid and a very experienced man. Bardsley's been in the game for 20 years and Marcus is a naive boy."
He continued: "I think it was a very, very good performance with a result which was short.
"They played very well as a team, and there were very good individual performances and we should score many more goals. The result shouldn't have been open until the last minute."
Rashford took to Twitter after the match to apologise for his conduct, writing: "Emotions got the better of me, I shouldn't of [sic] reacted like that. Sorry to everyone at the club and all the fans."
Mourinho started Marouane Fellaini as a deep-lying midfielder, as well as under-fire centre-backs Chris Smalling and Victor Lindelof, and he was impressed by the performances of all three, alongside in-form Luke Shaw.
"There were lots of good performances but Fellaini was fundamental, together with Smalling and Lindelof," he said. "Luke Shaw had a good performance and there were many others, too.
"We lost so many chances and Joe Hart made important saves. I don't remember such a comfortable performance as we've showed in the last two matches."
One of United's best chances in the second half came via a Paul Pogba penalty, which was saved by Hart, but Mourinho refused to criticise the France midfielder for his miss.
"I never blame a player for missing penalties," he said. "I blame the ones who refuse to go up there. Paul's scored some good ones for us."
Mourinho was again given a loud reception by the United fans, just as he was by a section of the Stretford End in Monday's 3-0 loss to Tottenham at Old Trafford, and he hailed the supporters for their backing.
"The quality of the performance and happiness of the team is an Old Trafford response – the way Old Trafford reacted to the boys in the Spurs defeat was important," he said to Sky Sports.
"The week was so good. I felt so bad leaving out people like Ander Herrera and Scott McTominay. It was heartbreaking to leave them out of the team.
"What Old Trafford did to the team after Spurs was fundamental to the happiness of the team. [On Sunday] the fans were fantastic from the first minute, but Old Trafford is a moment to remember."