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.
Cheltenham were on top, but Burton pulled one back in first-half stoppage-time.
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.
- Sports & Recreation
- Jamie Cureton
- Byron Harrison