Joel Gautschi gave the visitors the lead in the 27th minute, but Calum Blair tied things up late in the second period.
The Pirates looked to heading for a valuable point, but with four minutes to play Harry Ferguson silenced the home fans with a deserved winner.
Aberdeen Lynx are now just a point behind the Pirates after a comfortable 9-0 home victory over North Ayrshire Wild.
Iain Malcolm (3), Jordan Leyden (2) and Ross McIntosh (2) led the scoring for Lynx in front of a bumper crowd.
Dundee Comets are up to third in the table after a 5-3 win at home to Moray Typhoons.
John Dolan (2), Lewis Gold (2) and Kris Phillips got the goals for the Comets who are now four points behind Lynx with two games in hand.
Aberdeen Lynx 9-0 North Ayrshire Wild
Dundee Comets 5-3 Moray Typhoons
Paisley Pirates 1-2 Edinburgh Capitals SNL
(Image permission: Al Goold Photo)