The win moved United (63 points) to within a point of fourth-placed Manchester City ahead of the pair’s derby clash on Thursday and extended United’s unbeaten run in the Premier League to 23 matches.
After a week in which United lost top scorer Zlatan Ibrahimovic and Argentine defender Marcos Rojo to serious knee injuries, the return of Rooney to the side and to scoring form was a much need tonic.
Mourinho said was delighted with the way his players responded after their extra-time win against Anderlecht in the Europa League on Thursday.
Rooney, who last started a game on March 4, was given the start for only the second time in 2017 with Mourinho opting to start with in-form 19-year-old Marcus Rashford on the bench.
The 31-year-old England international could have scored as early as the ninth minute when he had a close-range shot blocked by Ben Mee and Burnley goalkeeper Tom Heaton was able to clear up.
Then Rooney was found on the left by Jesse Lingard but his lofted shot sailed just over the bar.
It was Rooney’s French strike partner Anthony Martial who opened the scoring in the 21st minute, breaking decisively from midfield and then finishing in style after swift exchange with Ander Herrera.
The goal will reportedly activate a clause in Martial’s contract where United must pay his former club Monaco around 8.5m pounds ($10.89m) when he reaches 25 goals for the club.
Rooney then made it 2-0, with his 251st goal for United, in the 39th minute, reacting well to poke home from close-range after Heaton had uncharacteristically failed to hold on to a low shot from Martial.
Burnley had plenty of possession in the second half but were unable to create any real chances apart from a diagonal shot from substitute Dan Agyei in the final moments.
United, happy to manage the game, had come close to a third in the latter stages when Paul Pogba fired just wide of Heaton’s right-hand post after a strong run from midfield.
The result leaves the Clarets in 15th place, five points above the relegation zone with four games remaining and facing some pressure after Hull City and Swansea recorded wins on Saturday.
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Former Liverpool footballer Christian Benteke bags one of his two goals for Crystal Palace against the Reds at Anfield REUTERS
Meanwhile at Anfield, two goals by Crystal Palace forward Christian Benteke, who Liverpool sold to their opponent earlier this season, cost the host the chance to put pressure on the Premier League's top two with a rare home defeat.
Philippe Coutinho had put Liverpool ahead in the 24th minute with a direct free kick.
The home side dominated but conceded an equaliser just before halftime when Benteke, sold by Juergen Klopp in August, scored from a cross by Yohan Cabaye. In the 74th minute the Belgian stooped to head in a corner and win the game.
Klopp's team stay third but are five points behind second-placed Tottenham Hotspur having played two games more.
Palace, who have recently beaten Chelsea and Arsenal, moved up four places to 12th and should now be safe from relegation after a sixth win in eight games.
Burnley 0 Man Utd 2
(Martial 21, Rooney 39)
(Coutinho 24) Palace 2
(Benteke 42, 74)
P W D L GD Pts
32 24 3 5 38 75
32 21 8 3 46 71
34 19 9 6 28 66
32 19 7 6 28 64
32 17 12 3 26 63
34 16 10 8 23 58