Martial started up front with Zlatan Ibrahimovic ruled out for the rest of the season due to a serious knee injury, and Jose Mourinho's decision to attack Burnley with the Frenchman's pace reaped dividends.
United broke off of a Burnley corner after 21 minutes, and Ander Herrera teed up Martial to finish past Tom Heaton from close range.
The Red Devils were two goals up six minutes from the interval as Paul Pogba stole possession and teed up Martial, with Heaton's save falling to captain Rooney to scuff home inside the six-yard box.
The win puts United just a point behind Pep Guardiola's Manchester City before Thursday night's derby clash at the Etihad Stadium, which could well prove to be a top-four shootout.
EPL Table | Premier League standings after Gameweek 34
In a game of half-chances, Manuel Lanzini's goalbound second-half effort was the closest any of the two sides came to finding a goal.
With the share of the spoils, the point takes Everton above Arsenal into sixth-place but the Gunners have three games in-hand over their Merseyside rivals. Meanwhile, West Ham edged closer to safety as they remain in 13th place, seven off the drop zone.
Elsewhere, Swansea and Hull record survival-boosting victories, while Bournemouth romped to victory against 10-man Middlesbrough.
Crystal Palace head to Anfield to face Liverpool on Sunday looking to cause another shock after recent wins over Chelsea and Arsenal.
Palace will have to do without loanee Mamadou Sakho, who is ineligible to face his parent club. Any positive result will all but secure Palace's Premier League status for another year.
Liverpool, though, will know a win will close the gap on second-placed Tottenham to two points having played two more games.