The Clarets returned to the top of the Championship with three goals in the second half.
Barnes opened his league account for the season with two goals, one of which came from Zaroury, as Rovers ended up on the wrong side of a local derby hammering.
It took the hosts until just before half-time to test Thomas Kaminski, who was in goal for the visitors.
Barnes put the visitors ahead on 55 minutes when he met Zaroury's left-wing cross and headed into the bottom right corner.
They doubled their lead on 74 minutes when Barnes' right foot volley on the turn was well saved by Kaminski, the ball ran loose and Zaroury drilled in a low right foot shot from the edge of the box.
Barnes scored his second goal of the game nine minutes from the end.
The Clarets were knocked off the top of the table by United's win on Saturday, a week after they were thrashed by the Blades.
After a defensive display from the visitors before the break, the Clarets came back with a three- goal margin of their own.
Just before the break, Belgian keeper Kaminski made two fine saves, first at point-blank range from on-loan Manchester City defender Taylor Harwood-Bellis, then tipping over Josh Cullen's goal, to change the course of the game.
Barnes got the better of Zaroury when he got the better of Hedges, and he scored the game winning goal.
Kaminski's fine save was pounced on by Zaroury after Barnes' well-struck volley.
Then Josh Brownhill skipped past Clinton Mola and pulled his cross back for Barnes, who calmly slipped a challenge to fired home the third, and stretch their advantage over third-place Rovers to five points.
A firework was let off near the end of the game, which caused its remnants to fall on the pitch. The players were not hurt by the falling debris.
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