The home side were on top for most of the afternoon and Alan Bennett saw a header from Billy Jones' free-kick hit the crossbar in the seventh minute.
Sido Jombati had a header from another Jones set piece hacked off the line and Steve Elliott nodded over from a good position.
Cheltenham went close again before half-time with Zebroski's close range effort hacked off the line following Marlon Pack's flick on from a Jones corner. Former Burton man Russell Penn's glancing header was saved by Mark Oxley after Marlon Pack's cross
Burton's best chance fell to Calvin Zola after a careless backpass from Kaid Mohamed in the 48th minute, but he snatched at the shot.
Zebroski won it for Cheltenham with four minutes left after Oxley spilt the ball following a scramble when Bennett saw a shot blocked on the line. Scott Brown saved well from Zola at the death to deny Burton a leveller.