Keeper Carl Wilkins was Abingdon's hero on his return to the club in the second round of the Berks & Bucks Senior Cup on Tuesday.
Wilkins saved two of Burnham's four spot-kicks after the teams had finished level at 2-2 in normal and extra time.
Burnham led through Dan Willmott at the break, but John Mills, an early substitute for the injured Jamie Brooks, levelled.
Darren Jones restored Burnham's lead before Ryan Curtin was brought down in the box with four minutes left and got up to score from the spot.
Abingdon Utd: Wilkins, Moss, Organ, Franklin, Woodley, Carbon, McCalmon, Herbert, J Brooks (Mills 10), Odhiambo (Hawtin 45), Curtin. Sub not used Hemmings.
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