The Bantams doubled their advantage in the 49th minute when Wells teed up James Hanson to fire home from the edge of the Burton penalty area.
However, the hosts halved the deficit seven minutes later when they earned a penalty. Garry Thompson brought down Jacques Maghoma, who himself stepped up to find the target from 12 yards.
That levelled the tie at 4-4, but inside two minutes Wells scored his second goal of the afternoon for Bradford. He collected Hanson's pass, held off Zander Diamond and shot home on the turn from close range.
That proved to be the final goal in the tie as Bradford qualified for the League Two playoff final.