Cheltenham, unbeaten in their previous six away matches, went ahead from the penalty spot in the 29th minute when Bradford defender Andrew Davies fouled lively winger Jermaine McGlashan and skipper Marlon Pack scored.
However, Bradford equalised with a penalty of their own in first-half stoppage time. Sido Jombati handled the ball as new loan signing Craig Forsyth went past him and Wells scored from the resulting spot-kick.
The Bermudian international then gave Bradford the lead in the 68th minute with a left-footed volley after substitute Zavon Hines had headed the ball across the goalmouth.
Left-back James Meredith then added a third in the 83rd minute with an angled drive after substitute Will Atkinson had played the ball into his path.
