The visitors had won six of their last eight games in all competitions going into the game and started impressively at Valley Parade, going two goals ahead within eight minutes through Andy Webster and Leon Clarke.
However, Wells scored his ninth and 10th league goals of the season to draw Bradford level before the half-hour mark, only for Callum Wilson to put Coventry back in front three minutes before the break.
As an away win looked on the cards, the Bermuda international coolly dispatched an equaliser from the spot in the final minute to ensure the spoils were shared.
Webster opened the scoring just two minutes in, firing home from inside the area after Jon McLaughlin had failed to deal with a lofted delivery, before the Bradford keeper saved Wilson's effort after he had outpaced Rory McArdle.
Clarke was on hand to apply the finish as the ball squirmed loose and double the visitors' advantage but, after 17 minutes, Wells pulled a goal back as he took advantage of sloppy Coventry defending to slot home.
The 23-year-old levelled matters 11 minutes later - acrobatically directing a James Hanson flick-on past Joe Murphy - before Wilson scored the fifth goal of the half to send Coventry in front at the break, cutting in from the right-hand side before beating Blair Adams and crashing home a superb strike.
The goals dried up somewhat in the second half but Wells continued to show his talents, hitting the bar with a free-kick after 55 minutes.
Jordan Clarke's handball in the final minute proved to be crucial, with the ensuing penalty duly converted by Wells to end Coventry's three-match winning run in the league.
