Rooney scored his 39th European goal as United crushed Feyenoord 4-0 on Thursday.
Rooney, 31, scored his 39th European goal as United crushed Feyenoord 4-0 on Thursday, thereby closing to within just one goal of Bobby Charlton's all-time club scoring record of 249 goals.
United need just a draw away to Ukrainians Zorya Luhansk in their final Group A game to progress and Mourinho says Rooney can be the inspiration behind the club's first success in the competition.
"It's an amazing achievement, obviously," Mourinho told reporters when asked to sum up Rooney's feat in surpassing Ruud van Nistelrooy as United's outright leading scorer in UEFA competitions.
"It will be even better if he can help to bring to the club the only competition that the club never won in its history, which is the Europa League.
"It's not the biggest competition, obviously, but it's a competition that Man United's history doesn't have. So let's try, let's push.
"I know that it's difficult. I know there are lots of matches in the knockout (phase).
"But let's try to qualify and if we qualify in Ukraine, then more matches to come and hopefully more goals for Rooney to score."
Rooney's performance followed a week of negative headlines after he supposedly 'gatecrashed' a late-night wedding party while on England duty.
Despite his milestone, the England captain missed out on the man-of-the-match award to his team-mate Henrikh Mkhitaryan, who marked his first start since September 10 with a fine display.
Mourinho said the Armenian playmaker had been left out of last weekend's 1-1 draw with Arsenal because he was not ready for the "pressure" of such a high-profile fixture.
While Mkhitaryan's direct running and neat footwork delighted the Old Trafford crowd, Mourinho said the former Borussia Dortmund star still had work to do.
"He just needs now to try to replicate this type of performance in the Premier League. The Premier League is different," said Mourinho, whose side entertain West Ham United on Sunday.
"He needs one more step. Obviously now confidence levels are higher.
"This performance gives him the right to believe that he can play the next match again and he needs to replicate this kind of performance in the Premier League, with more physicality, more aggression.
"He needs a little bit more of that, but the quality, we know why we bought him."
Rooney gave United a 35th-minute lead, starting the move in his own half and exchanging passes with Zlatan Ibrahimovic before coolly chipping Feyenoord goalkeeper Brad Jones.
Rooney teed up Juan Mata for United's second goal in the 69th minute and after an own goal by Jones, substitute Jesse Lingard sealed victory in stoppage time.
United leapfrogged Feyenoord to second place in Group A, a point below leaders Fenerbahce.
Feyenoord must now win at home to Fenerbahce in their final group game if they are to guarantee a spot in the round of 32.
"It's of course a result we didn't want," said beaten manager Giovanni van Bronckhorst.
"We could cope with Manchester for 70 minutes. The second goal was a big punch for us. We had a big chance to score the first goal.
"You know against Manchester, you don't get many chances, but we created one or two big chances to score the first goal. After 70 minutes, you could see the difference in quality."