On a night of third-round replays, this tie saw the sides going at it for the first time - after their initial attempt to play the game 10 days ago was scuppered by a frozen pitch.
Goals from Lukas Jutkiewicz and Kevin Long ultimately sent the Championship Blues through after Rovers, currently bottom of League One, had gone ahead through Ben Stevenson’s stunning effort.
Birmingham now face a trip to Blackburn in the fourth round as reward for their efforts in Nailsworth.
After collecting the ball in his own half, Elliott took advantage of the backpedalling Wolves defence before lashing a 25-yard effort high into the net, beyond the reach of an out-of-position Jose Sa.
Fabio Carvalho had a goal disallowed for offside but Elliott’s goal was enough to secure a 1-0 win for the defending champions in this replay, with their initial fixture ending in a contentious 2-2 draw.
Earlier this month, Toti Gomes’ strike which would have secured a Wolves win was dubiously ruled out amid a VAR failure, but there was no such drama this time around.
Liverpool’s reward for prevailing in the Midlands is a trip later this month to Brighton, where Jurgen Klopp’s side were handily beaten 3-0 in the league to leave the Merseysiders ninth in the table.
The teams shared a 1-1 draw in their first encounter and could not be split over another 90 minutes as Mark Sykes’ opener for the Robins was cancelled out by Swans substitute Ollie Cooper.
But Bell cut in from the left in extra time and fired past Andy Fisher in the Swansea goal to settle the replay and reward his club with a home tie against West Bromwich Albion in an all-Championship clash.
The Baggies ruthlessly exposed the 70-place gap between themselves and National League Chesterfield to prevail 4-0 at The Hawthorns.
John Swift scored the first then had a hand in the next two goals from Tom Rogic and Jake Livermore before substitute Jovan Malcolm headed in a fourth for West Brom, who were taken to a replay after a thrilling 3-3 draw at the Technique Stadium 10 days ago.
It was a 10th win in 12 games in all competitions for Carlos Corberan’s in-form side.
Thelo Aasgaard volleyed Wigan ahead within seconds of the second half starting but Cauley Woodrow scored at the second attempt five minutes later after goalkeeper Ben Amos had done well to initially parry away.
Extra-time loomed but there was to be one final twist, with the ball being played in from the Luton left and Adebayo firing home from close range to earn the Hatters a clash against Grimsby.