The astonishing finale left Palace bemoaning a sixth straight loss which has sucked them to the brink of the drop zone.
Palace looked to have secured the points when Christian Benteke scored six minutes from time to put the South Londoners 4-3 up after they had trailed 3-1 with 15 minutes remaining.
The visitors had drawn first blood on 19 minutes through Wilfried Zaha but Gylfi Sigurdsson levelled nine minutes before the break with a free kick before a brace of goals from Leroy Fer moved Swansea 3-1 clear at the Liberty Stadium.
Palace roared back with three goals in quick succession as James Tomkins netted following a corner, before Jack Cork put through his own goal and Benteke poached another.
But Llorente then floored them at the death to hand Swansea's US manager Bob Bradley, a surprise October replacement for the sacked Francesco Guidolin, a first win bonus.
Palace's abject recent form had left boss Alan Pardew under pressure at the club he took to FA Cup final appearances as player and then, last season, as manager.
They began the encounter just a point clear of the relegation zone and their late collapse means Swansea, who moved above Sunderland after the latter's loss at Liverpool, are now just two points behind Pardew's men, who are only out of the bottom three on goal difference.
Palace's losing run follows three successive September wins and it was similarly erratic form that cost Pardew fan support in his turbulent four years with Newcastle before quitting St James' park in January last year.
The South Londoners went into the Swansea game with the worst form across 2016 in the whole of the English league structure, having garnered just 0.71 average points per game.