Rashford scored United's opening goal with a cool finish and was a constant threat to the Chelsea defence with his pace as he started up-front alongside Jesse Lingard in the absence of rested Zlatan Ibrahimovic.
The 19-year-old, who has now scored in consecutive league matches for the first time since September, received a standing ovation when he was replaced by Ibrahimovic with seven minutes remaining, long after Ander Herrera had doubled his side's lead.
And Mourinho was thrilled with the display of his striker in the 2-0 victory, as well as highlighting the contributions of centre-backs Eric Bailly and Marcos Rojo.
"Rashford played fantastically well," Mourinho said.
"He also had a very good game at Stamford Bridge in the FA Cup. He had exactly the same chance to score then even when we were playing with 10 men in the second half because he is faster than their defenders and can get in behind them.
"This time he scored. It was early in the match and it gave the team more confidence and stability. It was very important. The kid played fantastic.
"But everybody fought really hard and did what they had to do. Eric Bailly and Marcos Rojo deserve a special mention because they play absolutely every game non-stop since the injuries of Chris Smalling and Phil Jones."
The result extended United's unbeaten run in the league to 22 matches and moved them up to fifth in the table, four points behind Manchester City with a game in hand.
Mourinho added at his post-match media conference: "Normally you win the title if you are 22 matches unbeaten and we are struggling to finish top four.
"Why? Everybody knows why because we lost too many points at home. But it's a great feeling to know that it's difficult for us to lose a match, people fight until the end.
"I am really happy with the team and with the result because Manchester City and Liverpool results left us in a position of having to win this match - not even a draw would have been a good result for us.
"I am really happy for us because we kept two windows open to try to play Champions League football.
"I cannot give up on the Premier League yet - we have to try. If one day we are in the Europa League and in the Premier League the distance is too big then we have to prioritise and nobody can criticise us if we do it in a different way in the last matches of the league.
"But while it is mathematically possible we will go with everything we have."