The visitors shocked their hosts by taking the lead after only eight minutes when a Scott Bannerman free-kick found an unmarked James Smith, who headed home from 10 yards.
It took Dumbarton only another four minutes to double their lead when Paul McQuilken made the most of some poor Thistle defending to shoot home from eight yards.
Thistle were desperate to get back into the match and they did so after 27 minutes when Gibson's pass found its way to Mark Roberts, whose 16-yard shot could only be parried by the Dumbarton keeper and the ball fell into the net.
Thistle's chances were given a further boost after 38 minutes when Ryan Borris was sent off for head-butting Sean Kilgannon.
The second half saw Thistle make the most of their numerical advantage, and following a concerted period of pressure they equalised after 61 minutes when Roberts completed a double by firing a 14-yard volley high into the net.
Thistle completed the turnaround after 69 minutes when Jukka Santala slid in at the back post to score from one yard after Dumbarton were unable to clear a corner.
Dumbarton had chances in the closing stages, and substitute Iain Russell should have done better than shoot wide after 83 minutes, but Thistle held on for a deserved victory.