Ellis Simms gave the Championship side the lead early in the second half and they were deservedly heading through to the last four before Rayan Ait-Nouri and Hugo Bueno turned the game around for Wolves.
However, Coventry snatched victory from the jaws of defeat, Simms equalising in the 97th minute before Haji Wright curled a perfect shot into the far corner of the net in the 10th minute of added-on time.
Both head coaches made two changes for this contest, Gary O'Neil unable to call upon Matheus Cunha despite a return to training and Jean-Ricner Bellegarde and Pedro also missing due to their respective injuries sustained against Fulham last weekend.
The only opening quarter-of-an-hour of the contest lacked goalmouth action, with Haji Wright having the best opportunity for Coventry as he lashed a shot just wide of the far post from inside the area.
Wolves also had concerns over a thigh injury for goalkeeper Jose Sa, which required treatment, while aside from one low threatening ball into the box, they did not cause Coventry any alarm.
Just before the half-hour mark, Wolves finally engineered a chance after good work from Nelson Semedo down the right, but each of Rayan Ait-Nouri and Mario Lemina could not force the ball through a pack of bodies and the opening went begging.
Semedo then fired straight at Brad Collins from 10 yards when he should have scored, before Sa made a strong low save to keep out Kasey Palmer's effort at the other end.
Ellis Simms somewhat missed from three yards when the ball was sent across the area, but it would have been ruled out for offside had he converted.
Simms turned provider soon afterwards, driving at the Wolves backline before teeing up Milan van Ewijk, but the wing-back's effort was diverted behind by Sa.
At this point, it was all Coventry, and Palmer should have done better when sending a side-footed volley straight at Sa, and Wolves were able to get to half time on level terms.
Upon the restart, both teams showed more ambition than they did at the start of the game, a rip-roaring run from Ait-Nouri ending with a shot going harmlessly wide with his weaker right foot.
However, Coventry would soon go in front. A free kick found Liam Kitching at the back post and his header back across goal was perfect for Simms to bundle home. VAR required several minutes to determine whether the ball had struck Simms's arm, replays proving to be inconclusive so the goal was allowed to stand.
O'Neil made the immediate decision to introduce Leon Chiwome for his debut, replacing the ineffective Nathan Fraser, and switch to a back four, yet it was Coventry who looked far more likely to double their lead.
Sa had to be alert to deny each of Wright (twice) and Josh Eccles in the same passage of play, before Palmer send a delicate effort agonisingly wide of the far post with the goalkeeper beaten all ends up.
Wolves finally responded, Joao Gomes drilling a 20-yard effort towards the top corner that was brilliantly tipped wide by Collins, before Ait-Nouri hit the post with a close-range header at the back post.
Back came Coventry with Simms unleashing a thunderous effort that was punched clear by Sa, before Callum O'Hare also saw Sa parry a long-range strike away from goal.
The Wolves goalkeeper was back in action yet again to keep out Kitching from point-blank range, but Wolves somehow found the leveller that their play did not warrant with seven minutes left, a cross being diverted into the path of Ait-Nouri who fired home from close range.
Ait-Nouri turned provider moments later with one of the assists of Wolves season. The Algeria galloped forward and in-field, drawing the attention of the defender, before threading a perfect through-ball to Bueno who calmly slotted into the bottom corner from 10 yards.
With nine minutes of added-on time to come, the job was far from complete for Wolves, and so it proved as Coventry made it 2-2. A cross from the left was flicked on by Bobby Thomas and Simms was on hand to head home from close range.
More to follow.