The former Burnley player clinched a visit to Sheffield United in the fourth round with the deciding goal 17 minutes from time.
Victory gave Blues a measure of revenge after losing their Premier League status at the expense of Wolves on the final day of last season.
Hughton's main priority is to try to regain City's top flight status at the first attempt - and they are on the fringe of the play-offs.
But his side had already showed their resilience in the Europa League this season and this was their 36th game in a crowded schedule.
Blues were worthy winners, creating the better chances while Curtis Davies impressed in defence.
The visitors knocked the ball around in confident fashion in the early stages and Wolves skipper Christophe Berra made a crucial interception to clear David Murphy's dangerous low centre, with Elliott behind him on the far post.
Sylvan Ebanks-Blake had the first shot on goal but City goalkeeper Colin Doyle was not extended with his low drive.
Ebanks-Blake then got in between two defenders to head narrowly wide from a testing Adam Hammill centre.
The Wolves striker just failed to connect with another quality cross, this time from overlapping right-back Matt Doherty.
Blues defender David Murphy and winger Jean Beausejour needed lengthy treatment after a clash of heads with both players going down the tunnel.
Wolves could not make their numerical advantage count and Adam Rooney had a good chance after 33 minutes.
A mistake by Wolves skipper Christophe Berra left the Blues striker with a clear run at goal - but left-back George Elokobi managed to get back and challenge Rooney as he prepared to shoot inside the box.
Blues enjoyed a spell of possession and again threatened to break the deadlock three minutes before the interval.
Jordon Mutch's inswinging free-kick picked out Curtis Davies whose glancing header forced a fine save from Dorus de Vries away to his left.
Blues finished the half on the attack and Jonathan Spector's cross-shot flew inches wide.
Davies continued to impress in the Birmingham defence at the start of the second period.
He got in a strong challenge on Ebanks-Blake as he attempted to run onto a Kevin Doyle pass, then blocked a shot on the turn from the latter.
De Vries turned aside a low, deflected attempt from Mutch.
At the other end Doherty's drive flicked off Blues defender Steven Caldwell and bounced up awkwardly but Colin Doyle was perfectly placed.
The Birmingham goalkeeper was also alert to cling onto Eggert Jonsson's shot through a ruck of players.
However, Elliott made the game's decisive breakthrough with 17 minutes left.
Elliott initially hit the post from a Mutch corner and, after Murphy's follow-up effort had been blocked, he poked the ball over the line from close range.
De Vries turned over a powerful drive from Beausejour after the out of touch Stephen Hunt was caught in possession.
Wolves sought an equaliser but Colin Doyle produced a superb double save to block bravely at the feet of Fletcher and then turn over the follow up attempt from Hunt in the 89th minute.