A tantalising replay now beckons in South Wales a week on Tuesday night.
League One basement side Stockport created the first decent chance after just two minutes. Full-back John Hardiker found space down the Swansea right but blasted an angled drive straight at goalkeeper Willy Gueret.
Kenny Jackett's side opened brightly too and soon after Lee Trundle twisted and turned before seeing his shot blocked by Hardiker.
Marc Goodfellow then let fly from 20 yards, but his shot was superbly finger-tipped over the bar by home keeper James Spencer.
Trundle went on another mazy run across the edge of the box on 24 minutes, and was unfortunate when his sweet shot rebounded back off the crossbar.
At the other end veteran Stuart Barlow went through one-on-one, but was thwarted by a smart Gueret save.
The game was building into a cracking cup tie, and next to go close was City's Roberto Martinez whose curling shot flew inches over the bar.
Substitute Barlow looked full of running having replaced the injured Warren Feeney early on, and his far-post header flew just wide on 34 minutes.
Just before the hour Barlow stooped to head Hardiker's cross goalbound, but again Gueret was there with a breathtaking one-handed save.
The Swans came closest to snatching victory late on when Paul Connor side-footed a shot wide from eight yards, but the visitors did survive a penalty appeal when Andrew Gurney looked to have hauled back Barlow.