The man in the middle wrongly pointed to the spot for a Clarets pen when their own player handled the ball, coming at a pivotal moment in the match when Swansea were well on top.
Burnley turned the game completely on its head and were on course for a smash-and-grab first away win of the season, only for the relegation-threatened Welsh outfit to show the character required to survive the drop by themselves turning things around in the topsy-turvy affair.
Martin Olsson scored 20 minutes from time to make it 2-2, before Fernando Llorente bagged his second of the afternoon in added time - nodding home from a Tom Carroll cross - to lift his side five points clear of the drop.
Swansea were denied by the frame of the goal on three occasions in the opening 45 minutes, the first coming early on when Alfie Mawson - a scorer in the Swans' last two games here - saw his header come back off the crossbar.
From the same phase of play, the ball was worked back into the defender's path and he was this time thwarted by the post to leave the hosts frustrated.
That theme of frustration was to continue for most of the afternoon, but the Welsh outfit were given a temporary reprieve when Llorente headed in goal number 10 of the league campaign on the 13-minute mark from a fine right-sided Leroy Fer cross.
A week on from being on the wrong end of a somewhat controversial handball call, Burnley this time benefited from the decision to award them a spot kick when their own player handled inside the box.
A cross into the area struck Sam Vokes on the arm, but Taylor and his officials did not see the incident clearly and instead pointed to the spot for Gray to drill down the middle.
Llorente saw his looped header hit the bar from one of the final acts of the first half, piling on further frustration in what had been a one-sided affair until that point.
Normal service was quick to resume after the restart, with Sigurdsson's shot saved off the line by Ben Mee and Llorente glancing another header inches past the far post.
Despite City's numerous opportunities it was to be Burnley who found the third goal of the afternoon, as Vokes knocked down a punt forward into the path of Gray, who swivelled and picked his spot to turn the game on its head.
Burnley were in front for less than 10 minutes, however, relinquishing their advantage when the lively Olsson got on the end of a Sigurdsson flick and rocketed home the equaliser from seven yards out - his second goal since joining in January.
Swansea looked like being the only winners as the final throes arrived, with Olsson volleying wide and Carroll seeing a shot kept off the line by Mee for a second time, but the triumphant Llorente moment did arrive at the death for three valuable points.
Swansea, who have now won three successive home games this season for the first time, remain in 16th place but can breathe a little easier at the bottom end of the division.