Despite the strong winds which put the game in jeopardy ahead of kick-off, the hosts took the lead after 17 minutes as Peter Crouch tapped in from close-range when Peter Odemwingie's effort rebounded off the post and kindly into his path.
Marko Arnautovic came close to doubling the scoreline when Michel Vorm was forced into action to tip his strike over the bar, before the goalkeeper then denied Jonathan Walters with a smart block from inside the area.
Swansea enjoyed a lot of the ball but real sights of goal were scarce in the opening 45 minutes as Jose Canas blazed Wayne Routledge's cross over the bar and Nathan Dyer fired wide.
Garry Monk's side came out from the break full of intent and they duly equalised after 52 minutes when Chico Flores guided Pablo Hernandez's pin-point cross past the flailing arms of Asmir Begovic with a cushioned header.
Wilfried Bony saw an effort heroically hooked off the line by Ryan Shawcross after Nathan Dyer had played him in, but both sides were made to settle for a point after a majorly scrappy second half.
Around 90 minutes before kick-off it appeared highly likely the game would be called off given the battering the stadium received from the winds.
Perched on high ground, Stoke's home venue is renowned as one of the bleakest in the country.
So given its exposure to the elements there was every chance it would fall by the wayside, as had two other top-flights matches at Manchester City and Everton.
But after surviving the two safety inspections, in fairness, the ferocity of the wind had eased by kick-off, although there were still gusts on occasion.
For the players it became a simple matter of keeping the ball on the ground, and in the opening exchanges it was Swansea who fared the better with such a task.
It was Stoke, though, who carved out the first chance, with Charlie Adam firing in an angled 20-yard drive in the 10th minute that was smartly smothered by Vorm down at his right-hand post.
Then came Crouch's opener, and it could have been 2-0 in the 20th minute but for Vorm tipping over the crossbar a dipping 20-yard effort from Arnautovic.
Swansea's best chance of the half came when Dyer flashed a 20-yard drive of his own across goal, with Begovic at full stretch.
Two minutes beforehand a petulant kick from Adam on Dyer, that earned the Scotland midfielder a yellow card, sparked a melee that also led to Swansea captain Ashley Williams having his name taken by referee Jonathan Moss.
In control at the end of the first half, though, Swansea were out of the blocks at the start of the second, with Bony firing just over the angle of bar and post from the edge of the area 90 seconds in.
Five minutes later and up popped Chico with the equaliser, and at that stage Stoke were undeniably second best all over the pitch.
That forced boss Mark Hughes into making a double swap soon after, with Steven N'Zonzi and Oussama Assaidi on for Arnautovic and Walters.
Four minutes after his introduction, N'Zonzi should have been on the scoresheet, but both he and Crouch were inches away from steering in an enticing cross from Adam.
Twelve minutes from time and Swansea thought they had the winner when a Bony shot was nudged goalwards by Begovic, only for captain Shawcross to steer it off the line.
Then in the 86th minute Begovic pulled off a stunning save to deny substitute Jonathan de Guzman from 22 yards.