Former Chelsea winger Scott Sinclair had given Swansea a 39th-minute lead when his left-footed volley looped over Petr Cech and into the net, putting the hosts on course for a repeat of their stunning win over Arsenal.
Chelsea dominated the second half but they looked set to suffer a sixth defeat of the season when Ashley Cole picked up a second booking late on.
However, with time running out, Jose Bosingwa's driven ball across the box from the right struck the unfortunate Taylor and beat Michel Vorm.
Chelsea, without captain John Terry due to a knee injury and with new signing Gary Cahill on the bench, had Branislav Ivanovic and David Luiz in central defence and they were thankful to the former for saving them twice during a sluggish opening from Andre Villas-Boas' side.
Ivanovic firstly denied Danny Graham a simple finish after a Steven Caulker header had rebounded to the striker before Chelsea threatened when Luiz ventured forward, with his low cross resulting in Daniel Sturridge firing narrowly past Vorm's near post.
Ivanovic was involved again when a long ball pumped towards Graham caused chaos in the visiting defence.
Cech could only clear the ball to Gylfi Sigurdsson, who skipped past the goalkeeper but saw Ivanovic block his goal-bound effort. Graham's follow-up attempt was kept out by Luiz and Joe Allen was denied by the recovering goalkeeper.
Having survived that scare Chelsea began to take a stranglehold on the game, dominating possession and pinning Swansea deep in their own half.
Ashley Williams made a terrific interception to cut out Fernando Torres' intended pass to an unmarked Sturridge, while Juan Mata fired wastefully wide of either post after more patient build-up.
Vorm was uncharacteristically hesitant in tipping a Oriol Romeu drive around his post, before Luiz failed to make the most of a free header from a Raul Meireles corner.
But after absorbing the pressure Swansea took the lead against the run of play as Sinclair netted against his former club.
Sigurdsson's free-kick was weakly headed away by Bosingwa, and a swing of Sinclair's left foot saw the ball float over Cech into the far corner, although the winger was restrained in his celebrations in front of the travelling support.
Stung by the concession of the goal so late in the first half, Chelsea completely dominated the opening exchanges of the second period.
Mata and Meireles prompted in midfield as Swansea defended ever deeper.
The visitors initially created little in the way of clear-cut chances, although they would have levelled had any blue-shirted attackers made the effort to follow in Cole's astute ball across the Swansea six-yard box.
Villas-Boas brought Michael Essien on for Romeu, and the move almost paid instant dividends as the Ghanaian blasted a fierce drive narrowly over Vorm's bar.
Bosingwa then teased his way to the byline but his hanging cross was headed clear by Angel Rangel, before another surge from the right-back saw Sturridge turn and poke wide.
Swansea were by now struggling to find their way out of their own half as their efforts to keep ball were stymied by Chelsea's pressing high up the pitch.
They summoned a rare attacking foray with seven minutes to go as substitutes Kemy Agustien and Luke Moore combined to give Nathan Dyer a sight of goal, but he dragged his shot across the face of goal.
Normal service was soon resumed as Bosingwa cut in and fired narrowly over, but Swansea were just about managing to handle everything Chelsea could throw at them.
Chelsea's frustrations were then compounded as Cole, who had been cautioned just before the interval, picked up a second booking for a rash tackle on Dyer and was dismissed.
But their pressure finally told as Bosingwa again tricked his way into the box and saw his driven effort strike Taylor to deceive Vorm and rescue a point.