The Minstermen’s Keith Lowe, James Berrett and Reece Thompson also converted, while home keeper Scott Flinders saved from Christopher Routis and Tony McMahon.
James Hanson and Josh Morris did successfully convert but that will be of little consolation for the Bantams after the score had ended 2-2 after extra-time.
Bradford had earlier opened the scoring in spectacular style after 21 minutes when Routis crashed in a powerful volley after being picked out at the far post by Alan Sheehan’s right-wing free-kick.
The home side’s best chance of a first-half equaliser fell to Thompson but he headed over inside the six-yard box.
Routis also volleyed a second chance wide from another Sheehan free-kick on the stroke of half-time.
The game came alive after the interval when Luke Summerfield converted a 49th-minute spot kick after Thompson went down in the penalty box under a Gary Liddle challenge.
Sheehan then skied over a 70th-minute penalty for the visitors after David Tuntonda felled Steven Davies.
York thought they had won the tie with five minutes of normal time remaining after a brilliant curling Berrett free-kick from 20 yards found Ben Williams’ top right-hand corner.
But James Hanson levelled in the fourth and final minute of stoppage time, firing a low drive past Flinders from 10 yards which took the tie to extra-time before the shootout.