United had scored just once in their last four league games at the stadium, with that goal coming from Lou Macari in April 1976.
Burnley also beat United 1-0 at home in 2009 and drew 0-0 at the beginning of the 2014-15 season.
Martial, who started the move before playing a neat one-two with Ander Herrera, calmly struck the ball past Burnley goalkeeper Tom Heaton after 21 minutes.
6 - Ander Herrera has provided six assists in the Premier League this term, as many as in his previous two campaigns combined. Inspired.— OptaJoe (@OptaJoe) April 23, 2017
43 - Man Utd have scored their first PL goal of the season against Burnley, with what was their 43rd shot against the Clarets. Peppered.— OptaJoe (@OptaJoe) April 23, 2017
The goal also marked the first goal against the Clarets in the Premier League this season, having drawn the reverse fixture 0-0 earlier in the campaign.
Martial's goal was the club's 43rd shot, however, with Zlatan Ibrahimovic wasting a host of opportunities at Old Trafford in October.
United move to just one point behind rivals Manchester City with a victory against Burnley, just days ahead of the Manchester derby at the Etihad Stadium on April 27.