The England international came on as a second-half substitute against the Clarets but it proved to be a short spell on the pitch as eight minutes later he was given his marching orders by referee John Moss.
Rashford has only started one of Uniteds four league matches this season and would have been eager to have made an impact as he bids to come in from the cold.
However, he reacted to a kick from Burnley full-back Phil Bardsley and left his head in on the former United defender, with Mourinho branding him as naive.
I dont know what happened. I would say it was naive, Mourinho said after the win.
It was a kid and a very experienced man. Bardsleys been in the game for 20 years and Marcus is a naive boy.
Rashford was quick to apologise to his manager and the clubs supporters after the match, taking to Twitter to vent his frustration at his behaviour.
Emotions got the better of me, I shouldnt of [sic] reacted like that, he wrote.
Sorry to everyone at the club and all the fans.
Goals from Romelu Lukaku wrapped up a much needed three points for the Red Devils, who are languishing in mid table after a slow start to the season.
Paul Pogba missed a penalty before Rashford was sent off, but they held on for a comfortable win.
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