Thistle dominated the early exchanges and they were awarded a spot-kick after 14 minutes when Andrew Considine rashly tripped Mark Roberts as the forward ran through on goal.
Referee Stuart Dougal had no hesitation in pointing to the spot and Considine was lucky to escape without any punishment despite being the last defender. Roberts stepped up to take the kick himself, but he sent the penalty crashing off Jamie Langfield's right-hand post.
The Jags controlled proceedings following the penalty miss and Liam Buchanan should have netted with a free header, before sending a long-range effort whistling over the bar seconds later.
Roberts then went close to firing the Jags in front with an angled shot that was parried clear by Langfield.
But the Dons eventually awoke from their slumber, punishing Partick for their profligacy in front of goal as former Jag Derek Young fired the Dons in front minutes before the interval.
Barry Nicholson saw his effort well parried by Thistle keeper Craig Hinchcliffe and as the ball rebounded to Young on the edge of the area, he fired home a sweet right-foot effort through a sea of legs.
Considine put the result beyond doubt when he doubled the Dons' advantage after 64 minutes.
Jamie Smith floated a corner kick to the back post where Considine was on hand to knock the ball home from close range.