The striker converted Charlie Taylor's cross in the 38th minute to net his 10th goal of the campaign and lift Burnley up to ninth in the Premier League.
Sean Dyche's side have now lost just once in 11 games, claiming 10 points from their last four fixtures, and kept a 14th clean sheet of the season.
Here's how we scored the players' performances...
Here's how we scored the players' performances...
. Nick Pope 9
A dropped catch in the first half aside, the England international goalkeeper was outstanding. Denied Soucek from close range, kept both Antonio and Yarmolenko out and made an excellent save to deny Haller in the second half.
. Phil Bardsley 7
Caught out a couple of times in the first half with balls over the top and in behind, but recovered well and was far more assured after the break. Fought for every ball and supplied a peach of a cross for Wood's disallowed finish.
. James Tarkowski 9
Another phenomenal display from Tarkowski at the heart of the Burnley defence. Starred in yet another clean sheet. Made an important block to deny Yarmolenko in the first half and won every other duel he contested, whether in the air or on the ground.
5. Charlie Taylor 8
Influential at both ends of the pitch. Kept busy defensively in both halves, working hard to limit crossing opportunities, and composed on the ball when carrying it out from the back. Supplied a superb cross for Rodriguez to convert the winner.
6. Erik Pieters 7
Sensible in possession, using the ball wisely when getting on it, and was disciplined without the ball to retain the side's structure down that right hand side. Dropped between the lines centrally to add another layer of protection when needed.
7. Josh Brownhill 8
Took one for the team right at the end as the Clarets fought tooth and nail to protect their advantage. Excelled in the middle of the park, controlling the play, assured in possession, controlling it well and switched play with some lovely weighted passes.
8. Ashley Westwood 8
Westwood was a running machine in the middle of the park. Closed the play well, fracturing United's attempts to break through the middle, and was always on hand to get a toe to the ball or put his foot through it on the edge of the box.