Luciano Narsingh's strike on the stroke of half-time put the Premier League's bottom club ahead but the underdogs responded with gusto and Stead scored the equaliser just after the hour mark.
With Swansea fighting to preserve their top flight status and Notts County battling for automatic promotion from League Two, a replay is probably what neither wanted but the two teams will do it all again at the Liberty Stadium next month.
Carlos Carvalhal made eight changes to the team that started the impressive win against Liverpool. Monday night matchwinner Alfie Mawson, fellow defender Mike van der Hoorn and midfielder Leroy Fer were the only players to keep their places.
Notts County made five changes, with former Newcastle striker Shola Ameobi the most notable name recalled.
The hosts came within a whisker of taking an early lead. Jorge Grant played in Stead down the right-hand side of the penalty area and Elliott Hewitt was inches away from turning in the low cross as he slid in at the far post.
Swansea took a while to find their feet and were restricted to a couple of long-range efforts from Van der Hoorn and Tammy Abraham, both of which worked Adam Collin without troubling the home goalkeeper.
However, the Premier League team began to impose themselves around the half-hour mark. A well-worked free-kick created space for Renato Sanches inside the penalty area but from his cross Van der Hoorn could only tamely head the ball straight into Collin's arms.
Wilfried Bony should have done better when Narsingh's cross found him unmarked six yards out but he powered his header straight at Collin, who showed good reflexes to keep it out.
Swansea made their pressure tell in the final minute of the first half. Substitute Ki Sung-yeung, who came on after Sanches hobbled off in the 32nd minute, rolled the ball into the path of the overlapping Narsingh down the right side of the penalty area and he beat the goalkeeper with a low shot into the far corner of the net.
Notts County responded brilliantly and enjoyed their best spell of the match at the beginning of the second half.
Grant went close with a free-kick and shortly afterwards saw a rising effort narrowly clear the crossbar. The on-loan Nottingham Forest midfielder was in the thick of the action and sent a header agonisingly wide.
He might have been unable to find the net himself but it was Grant's low cross across the face of the goal which was turned in at the far post by Stead for the equalising goal after 62 minutes.
The League Two side had spent a lot of energy getting back into the game and Swansea's superior fitness was evident in the closing stages, with Kevin Nolan's men finding it increasingly difficult to get out of their own half.
Collin was only really tested though with 10 minutes remaining when he had to get down by his right-hand post to push away a strike from Ki, although Bony bounced a header just wide right at the end but Notts County held on.
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