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...
5. Erik Pieters 8
Victim of an over-exuberant challenge from Drmic, which led to a second red for the hosts, but unaffected. Allowed to get on the ball more after the break, and take up more advanced positions, as he tested Krul and claimed an assist.
6. Johann Berg Gudmundsson 7
Tracked Hernandez and Lewis well to sure up that right hand side and always looking to break between the lines when getting on the ball. Provided the assist for Wood's 13th goal of the season and then replaced by Brady on the hour.
7. Josh Brownhill 7
Squeezed the play well in the first half to negate the threats of Cantwell and Buendia, who were unable to get on the ball in key positions. Covered plenty of ground, switched play well and kept things ticking over in the second half.
8. Ashley Westwood 7
Always in the thick of things, with Buendia reacting off the ball, and constantly getting tight to his men to prevent them turning on the ball. Slowed things up for Norwich, who struggled to penetrate, and worked well with Brownhill.