Jamie Cureton put Cheltenham in front on his debut, pouncing after Burton goalkeeper Jordan Pickford was deceived by a long clearance by his opposite number Scott Brown.
But there was a price to pay with Cureton taken off on a stretcher after being clattered as he guided the ball into the net.
His replacement Byron Harrison doubled Cheltenham's advantage in the 32nd minute, firing low into the bottom corner after Jermaine McGlashan's shot was blocked by Marcus Holness.
Cheltenham were on top, but Burton pulled one back in first-half stoppage-time.
Russell Penn was adjudged to have handled a cross into the box after Rory Delap's long throw caused problems and Billy Kee converted from the spot.
And Burton pulled level 11 minutes into the second half when Delap marked his debut with a goal from Alex MacDonald's corner.
Sam Deering hit the bar for Cheltenham, while Pickford denied Harrison a second with a fine stop as honours between last season's beaten play-off semi-finalists finished even.