Colchester were reduced to 10 men when substitute defender Daniel Pappoe was sent in the first minute of stoppage time.
Pappoe was shown the red card for bringing down Bradford substitute Mark Yeates after only four minutes on the pitch.
Wells put Bradford in front after 14 minutes, the ball falling into his path from an attempted clearance by David Wright and he spun past Tom Eastman to slot the ball into the net.
But the lively visitors crowned a strong spell of pressure with two goals in four minutes.
Jeffrey Monakana equalised in the 28th minute with an angled shot into the far corner after taking a pass from Craig Eastmond and Colchester went ahead four minutes later when Monakana's through-ball split the Bradford defence and Jabo Ibehre fired past advancing goalkeeper Jon McLaughlin.
Wells equalised with his second goal of the match in the 52nd minute when his looping back header from left-back James Meredith's cross deceived U's goalkeeper Sam Walker as the ball floated in the far corner, while both sides went close in the closing stages.