The Clarets, who also kept a 15th clean sheet of the season against Norwich City at Carrow Road, moved to within a point of seventh place Spurs with just one game remaining.
Daniel Farke's Canaries were reduced to 10 men before the break, but the visitors showed their professionalism in the second half to see the game out comfortably.
The result, which stretched City's wait for a goal to 830 minutes, saw goalkeeper Nick Pope restored to pole position in the race for the Golden Glove.
Here are the ratings...
9. Dwight McNeil 8
It promised to be an intriguing battle between McNeil and Aarons, with the former starting to get on top just before City lost their discipline. Moved into a more central position after the break in a bid to breach the home side's resilience and resolve.
10. Jay Rodriguez 8
Will be disappointed not to have found himself on the scoresheet, but, as always, he was a nuisance for the opposition's defence. Continued to drop into different areas of the pitch to keep City guessing and unfortunate not to score with a second half flick.
11. Chris Wood 7
Set a new club record with a 13th Premier League goal of the season. It wasn't his cleanest finish, but he won't care. Struggled to make an impact prior to that and should have added to his tally in the second half, especially from Westwood's pass.
12. Robbie Brady 6
Managed to get a crucial half-an-hour in the tank against his former club as he continues to work his way back from injury. Always looking to swing the ball in behind the opposition defence.