Cheltenham nearly took an early lead but Andre Blackman was in the right place at the right time to head clear off the line before Eisa hit the rebound well over the crossbar.
Jamie Stephens conceded an indirect free-kick in his own area when his attempt to clear a backpass flew up in the air and he caught the ball. Jamal Campbell-Ryce blocked the effort behind for a corner kick.
Dawson opened the scoring in the 32nd minute with a finish at the far post after getting on the end of Mohamed Eisa’s was given space to deliver a dangerous ball across the face of goal.
The visitors doubled their lead in the 6 minutes later when Winchester was allowed to weave his way through the Barnet back line to score.
The Bees best chance to get back into the game came midway through the second half when they went racing down on goal, two-on-one. Shaquile Coulthirst opted to shoot instead of playing John Akinde over to his right through on goal, and it was blocked by Jamie Grimes.
It could have been worse for Barnet but Will Boyle’s free header went well wide before the tricky Eisa curled a shot narrowly past the post.
Line-up: Stephens, Brindley (Vilhete 82′), Clough (Akinola 45′), Santos, Nelson (c), Watson, Akinde, Campbell-Ryce, Coulthirst (Nicholls 79′), J. Taylor, Blackman.
Unused subs: Ross, Tutonda, Fonguck, Sweeney
Attendance: 1487 (198 Cheltenham fans)
Amber Big Draw: 05616