Former Owls boss Carlos Carvalhal watched his Swansea side have the better of the game but the Championship outfit were well worth a share of the spoils, which has resulted in a replay being scheduled to decide who will progress through to the last eight of the competition.
Wednesday dominated the opening stages of the game in Yorkshire but their only clear opportunity came in the second minute, when Adam Reach was given the space to surge into the penalty area before having a shot across goal tipped wide by Kristoffer Nordfeldt.
Atdhe Nuhiu also failed to make the most of a tough headed opportunity from the penalty spot, and Wednesday's failure to do more in the final third allowed Swansea to come back into the match.
Kyle Naughton - a late replacement for Alfie Mawson who was injured during the warm-up - saw a deflected effort go wide of the far post before the defender's cross almost led to the visitors taking the lead.
Wednesday finished the first half on top but like at the start of the match, the Championship outfit were struggling to create clear-cut opportunities in the final third.
The first 15 minutes of the second half passed by without incident, with only a speculative header from Reach representing a shot from either team.
Carvalhal made the decision to introduce Jordan Ayew from the bench and within five minutes of his introduction, the attacker got two shots away from distance, both of which were comfortably saved by Dawson.
Nuhiu was proving to be a frequent nuisance for Wednesday in attack but his best chance of the game took until the 78th minute, when his low shot towards the near post was claimed at the second opportunity by Nordfeldt.
Chances remained few and far between during the final 10 minutes and neither club could argue with a second fixture being required in South Wales.
SHEFFIELD WEDNESDAY (5-4-1): Dawson; Hunt, Venanico, Jones, Pudil, Fox; Boyd, Butterfield (Pelupessy 70'), Reach, Wallace (Joao 80'); Nuhiu
SWANSEA CITY (3-4-3): Nordfeldt; Naughton, Bartley, Van Der Hoorn; Roberts, Carroll, Ki, Olsson (Clucas 79'); Narsingh (Dyer 64'), Routledge, Abraham (Ayew 67')