The Clarets' equaliser came in dramatic fashion, Andre Gray dispatching a penalty that was given because Gray’s strike partner Sam Vokes handled in the area, with referee Anthony Taylor believing it was a Swansea hand.
Llorente scored a powerful header before the penalty to put the Swans ahead.
In the second-half Gray doubled his tally, finishing after Vokes' knock-down, before Martin Olsson powered a shot past Paul Robinson.
Llorente then won the game in stoppage time, heading home from a deep cross by Tom Carroll to hand Swansea a huge three points under Paul Clement as they put themselves five points away from the relegation zone.
Action-packed first half
The first effort came after 10 minutes as Alfie Mawson struck the woodwork twice. His first came back off the bar as he met Sigurdsson’s free-kick with a header and the ball fell back to him but his right-footed strike hit the post.
Llorente then netted the opener just two minutes later. Kyle Naughton launched a long ball to Fer in the corner, he managed to get a cross away that Llorente met with power at the back post.
Burnley’s first chance came shortly after, as Jeff Hendrick, who was coached at Derby County by Swansea manager Paul Clement, tested Lukasz Fabianski from range but the goalkeeper was equal to his effort.
The Clarets came out of their shell, as Gray backed up on Federico Fernandez, his shot was deflected which forced an outstanding save by Fabianski.
The penalty incident came a minute later. Off a corner there were a number of bodies inside the penalty area and the ball fizzed past them all before striking Vokes’ arm. He clearly handled the ball, but a penalty was given to Burnley.
Recent claims that English football needs more video technology will only be bolstered by the events that occurred at the Liberty Stadium this afternoon.
Gray then slotted home the penalty straight down the middle, just the second goal Burnley have scored in the opening half of their away games this season.
Olsson then had three chances all within five minutes, two of which were well saved by 37-year-old Robinson, and the other was blocked.
Swansea responded very well to the equaliser, as Jack Cork pushed through the midfield, teed Sigurdsson up just outside the area but he dragged his shot just wide of the goal.
The action continued up to half time, as Llorente hit the bar with a looping header after some good work down the left flank by Olsson and Sigurdsson.
The Icelandic midfielder then had his free-kick saved by Robinson on the stroke of half time.
Second-half following first 45's lead
Just two minutes into the second half, Sigurdsson curled an effort towards the bar post but Ben Mee was able to get back and clear off the line.
Llorente nearly doubled his tally with a header, but enough pressure from Mee put him off and his effort went wide.
Sigurdsson met Tom Carroll's cross after an hour had been played, he couldn't get his flick on target to worry Robinson in net though.
Against the run of play, Gray doubled his tally for the day. Vokes outjumped Fernandez and Gray was able to slot home after some poor marking.
Vokes almost got his goal shortly later, as Gray tried to return the favour and tee up his strike partner but the Welshman couldn't turn the ball home just a few yards out from goal.
Swansea then equalised after a sublime piece of skill from Sigurdsson found Olsson, a threat all afternoon, who finished with power.
Carroll then saw his injury-time volley saved off the line once again by Mee as Swansea pushed for a winner that finally came.
Clement was running down the touchline to celebrate with his side as Llorente towered above Burnley's defence to head past Robinson.
The result sees Swansea stay 16th on 27 points, while Burnley drop to 12th.