The match hinged on a remarkable 60-second period midway through the second half at the Liberty Stadium - a minute that started with a missed penalty from Marko Arnautovic and ended with a deflected Tom Carroll strike that wrapped up valuable three points.
Llorente's first goal in five Premier League games had put the hosts ahead early in Saturday's clash, although Swansea failed to add to the Spaniard's 12th goal of the campaign before the break.
It looked as though Mark Hughes' side, who were poor throughout, would be handed a reprieve when Xherdan Shaqiri was bundled over in the box after 69 minutes, only for Arnautovic to thump the resulting spot-kick over the crossbar.
And Stoke's misery was compounded exactly a minute later, as Carroll's long-range effort deflected in off former Swansea man Joe Allen.
1:00 - There was exactly one minute between Marko Arnautovic's missed penalty & Tom Carroll's goal for @SwansOfficial. Emotions.— OptaJoe (@OptaJoe) April 22, 2017
Casual play at the back almost cost Stoke early on, but Bruno Martins Indi recovered to prevent Gylfi Sigurdsson cutting the ball back to Llorente in the six-yard box.
Sigurdsson, who has provided more assists from set plays than any other Premier League player this term, did manage to tee up Llorente 11 minutes in, though.
The unmarked former Juventus striker made no mistake with a thumping header to the right of Jack Butland after latching on to Sigurdsson's out-swinging corner.
Butland, making his first start since March 2016, almost gifted Swansea a second soon after but Leroy Fer could not take advantage.
Fer took a heavy knock from the England goalkeeper for his troubles, and the Dutchman was soon replaced by Ki Sung-yueng when it became clear he could not carry on.
In a flurry of activity on the stroke of half-time, Ki's superb throughball played in Jordan Ayew, whose cross-cum-shot flashed across the face of goal, before Saido Berahino headed straight at Lukasz Fabianski at the other end.
Butland's nervy return to action continued just after the hour mark, Stoke's goalkeeper fumbling Sigurdsson's free-kick into the path of Alfie Mawson, who could not apply the finish.
Stoke were presented with an ideal opportunity to restore parity moments later, though, when Federico Fernandez clumsily brought down Shaqiri.
Arnautovic, who netted a stunner against Hull last time out, stepped up to take the penalty but sent his effort soaring into the stand behind the goal.
Swansea wasted no time in making Stoke pay, Carroll unleashing an ambitious shot that looped into the top-left corner after taking a wicked deflection off Allen.
Carroll could well have added a third with another venomous strike soon after, but Butland pulled out a top-class save to ensure Stoke avoided any further damage.
Key Opta stats:
- The Swans have won four of their last six home Premier League games, having won just two of their first 11 at the Liberty Stadium this season.
- There were just 60 seconds between Marko Arnautovic’s missed penalty for Stoke and Tom Carroll’s goal for Swansea.
- No player has scored more headed goals than Fernando Llorente in the Premier League this season (6 - level with Romelu Lukaku and Michail Antonio).
- Only Kevin De Bruyne (13) has more Premier League assists than Gyfli Sigurdsson this season (12 – level with Eriksen), while the Icelandic international has provided more from set plays than any other player in the competition (7).
- Stoke have failed to score in six successive away league games for the first time since August 1997 (a run of nine).
- Tom Carroll’s goal was his first in the Premier League since Boxing Day 2015, when he scored for Spurs in a 3-0 win against Norwich.
- Paul Clement has the best win percentage of any Swansea manager in the Premier League (40 per cent - six wins from 15 games).