The Championship outfit forced a number of fine first-half saves from Swansea 'keeper Kristoffer Nordfeldt but saw Rúben Vinagre sent off for catching Nathan Dyer high with his studs before half-time.
Martin Olsson's free-kick crashed against the crossbar as they looked to make the most of their numerical advantage but it was 10-vs-10 after 67 minutes when Swansea's Leroy Fer was given his marching orders for a cynical trip on Hélder Costa as Wolves looked to counter.
But despite an open contest, and the introduction of new signing Rafa Mir late on for Wolves, neither side could find the all-important goal and will have to go head-to-head again at the Liberty Stadium later this month.
Wolves the better side until Vinagre red
The hosts started brightly as both sides jostled for control and Bright Enobakhare forced a strong stop from Nordfeldt, while Morgan Gibbs-White was also denied by the Swede when one-on-one, both inside the first 10 minutes.
While Swansea grew into the game to enjoy better spells of possession, they struggled to trouble Wolves - who looked the far likelier side to produce the breakthrough goal. They went close again when Costa forced another sprawling Nordfeldt save.
Only Renato Sanches' harmless long-range effort brought Will Norris into action inside the first half-an-hour with Wolves justifiably living up to their billing as pre-match favourites despite being a division lower.
And Nuno Espírito Santo's side should really have taken the lead after Alfred N'Diaye's header deflected out to Enobakhare but he struck his first-time half-volley wide of goal with Nordfeldt scrambling to get back to his line.
Sanches was forced off with an apparent hamstring injury soon after before the Swans went close through Martin Olsson, whose free-kick on the edge of the box forced Norris to tip on to the crossbar and over.
Just two minutes later another Portuguese player's afternoon ended early when Vinagre stretched to challenge Nathan Dyer but caught the winger with a nasty challenge just below his knee and rightly saw a straight red card from referee Anthony Taylor.
Wolves restored their grip on the game after the interval but failed to create chances with the same tempo, substitute Barry Douglas curling a left-footed free-kick into the wall.
Swansea countered immediately from that with Wilfried Bony's powerful effort testing the reflexes of Norris, who was also equal to Olsson's tame low effort a minute later.
Fer sent off for Swans in slower second-half
But Wolves - looking to avoid a replay as they introduced Ivan Cavaleiro on 64 minutes - were dealt a huge boost when Fer was dismissed for tripping Costa, evening up the numbers.
Kortney Hause should have tested Nordfeldt from a Douglas free-kick but headed over when left in space at the back post as Wolves searched for a winner.
Mike van der Hoorn's crucial intervention prevented debutant Mir, signed from Valencia in mid-week for £1.5 million, from making more of a late chance while Nordfeldt had to push Mir's header from the resulting corner away from danger.
But while neither team was short in endeavour, they could not produce a goal to win the game and will have to do it all over again - this time in the middle of the week, later in January.
Anthony Taylor takes centre stage
Vinagre's red card soured the mood inside Molineux with the home fans deeming the decision harsh, jeering Dyer's every touch from there on out and letting the official know of their opinions on his way down the tunnel at half-time.
But suggestions Taylor's first red card was harsh were misplaced, with the full-back landing his studs high on Dyer's leg - a challenge which the winger was perhaps lucky to emerge unscathed from given his leg was planted to the turf.
Instead it was the visitors who have the right to feel let down by the referee for his contentious straight red card for Fer.
The midfielder certainly warranted a booking for his cynical challenge on Costa, who had broken free of him in the middle, but whether red was the right card is a completely different debate.
It was Wolves who had the better chances and probably should have won the game, but Swansea will feel they might have taken the ascendancy late on had Fer not been bizarrely sent off.
Bright showing from new boy Mir
Wolves' latest recruit, signed to ease the pressure on top scorer Léo Bonatini, may have only been afforded 14 minutes - plus four minutes of stoppage time - but made a positive first impression on the 21,000 Wolves fans who braved the freezing Black Country temperatures.
If not for van der Hoorn's superbly-timed block inside the box, he might have been celebrating a first goal for the club inside his first 10 minutes in a black and amber shirt.
It was the 20-year-old Spaniard who also rose to meet the following corner with his header close to finding the top corner if not for Man of the Match Nordfeldt's fingertip save.
And for good measure Mir - who scored 15 goals in 19 appearances for Valencia's B team this season - also headed just wide from a Matt Doherty cross soon after, although he was flagged offside.
His energy, running and movement re-energised Wolves' attack and while it did not lead to a goal, his performance was a huge positive. Having insisted the forward will add "good options" to his team, Nuno will most definitely be encouraged by Mir's first display.
Wolves are cut out for the top-flight
Swansea boss Carlos Carvalhal claimed pre-match that were Wolves of Premier League status right now, they would be a mid-table team. On this evidence it's hard to argue with him, even if they failed to win the game.
Wolves were comfortably superior in the first half, until the final five minutes that followed Vinagre's red card, and despite being down to 10, regained their control in the second period.
They might have made more of their opportunities - particularly in the first 30 minutes - but their dominance, even despite making six changes and resting the likes of Rúben Neves and Diogo Jota, suggested they are capable of putting together a good run in this competition if they win the replay.
Chances did not flow as freely in the second-half, naturally, but still Wolves were the better side for large periods - Swansea's threat increasing but not to the point where Nuno's men ever looked worried.
Swansea may have been the side with the Premier League standing, but it was Wolves who played with the authority of a top-flight team on this occasion - and this was a weakened side who were not at their best.
They'll hope to do so again when they face off again in South Wales.
This is now Southampton, Manchester City and Swansea who they have gone toe-to-toe with this season - playing well against all despite mix-and-match line-ups.
It will serve them as a very good learning curve given they are destined to go up against the country's best teams next season.