Burnley's main threat came from Kyle Lafferty, but when he did get a shot on target in the closing stages, David Marshall made a fine diving save.
The Canaries kicked off attacking the Barclay-End goal and Dublin opened the scoring when he steered Darren Huckerby's low cross from the right, inside the far post with a low right-foot tap-in from five yards.
Another Huckerby cross came close to providing another scoring chance, but the Clarets weathered the storm and Lafferty brought a save from Marshall with an angled shot.
In the 18th minute Dublin's pass released Darel Russell, but his scuffed shot went straight to Brian Jensen.
Joey Gudjonsson brought a good save from Marshall with a long-range drive.
In the 40th minute the lively Huckerby cut in from the left-wing and fired over with a crisp right-foot shot.
At the other end a mistake by Jon Otsemobor let in Lafferty, but he fired wide and Norwich went in at the break with the lead intact.
Eight minutes into the second half, determined play down the Norwich right by Matty Pattison set up Jamie Cureton but he screwed his shot wide.
The home side continued to do most of the attacking without coming up with a second goal until the 90th minute when substitute Evans collected a pass from Huckerby.
He beat his marker and fired an angled left-foot shot from ten yards, beating Jensen low to his left to send the jubilant Norwich fans home in eager anticipation of next week's local derby with Ipswich Town.