The Bees' shot-stopper pulled off countless saves to deny an all-out Swans attack on his goal during the second half.
But just as it looked like Swansea wouldn't add to Bayo Akinfenwa's opener after it was cancelled out by Andrew Frampton, Robinson struck with a curling free-kick just four minutes from time.
And the result leaves the Welshmen just one point off second-placed Brentford and with no defeat in ten games.
Akinfenwa had already caused the Brentford defence plenty of problems before the opener, twice pressurising Michael Turner into mistakes inside the first 15 minutes.
And after Lloyd Owusu failed to make the most of his 19th minute chance when he was allowed to test Willy Gueret in the Swansea goal, the hosts were in front 60 seconds later.
After some good wing work from Robinson, Sam Ricketts crossed and as the Brentford backline failed to clear Leon Britton's header, Akinfenwa poked home.
But the lead would last just ten minutes as Frampton headed home unmarked from a Jay Tabb corner on the half-hour mark to leave it all square at the break.
Nelson had to be at full stretch to stop Lee Trundle curling home a 41st minute free-kick to keep it level at the break.
But it was the visitors who would next go closest to scoring, confusion between Alan Tate and Keith Lowe in Swansea's defence sending Paul Brooker clear on 52 minutes, but Gueret was equal to his effort.
Sub Rory Fallon then thought he had found the perfect way to open his Swansea account when he shot from long-range on 60 minutes, only for Nelson to tip his effort on to the bar.
Nelson's work wasn't done as he stopped further efforts from Fallon and Trundle as well as a double from Adrian Forbes to keep the Swans frustrated before Robinson won it late on.