Earlier, it had looked as if Thistle might get their first points of the season, for Kris Doolan scraped the bar with a 15-yard shot in the 15th minute.
Ten minutes before the turn Iain Flannigan finally gave Thistle the lead they deserved when he took advantage of a mix-up in the home defence between Stephen McKenna and David Lilley. Flannigan turned and shot home from 20 yards, giving David Hutton no chance.
The Doonhamers got their reward for some strong pressure in the 76th minute. Johnston and Paul Burns combined well and Johnston was left clear on the edge of the box to shoot home for the equaliser.
They struck in the 87th-minute as Quinn rounded a couple of Thistle defenders before lashing the ball into the net for a great winner from the edge of the box.
Queens left it late and were extremely fortunate to win three points as Thistle dominated for long periods after taking full advantage of the defensive lapse which led to their opening goal.
Kenny Brannigan was naturally happy with the final result but admitted, "We were a bit lucky to get all three points but we had to fight really hard for them.
"In the early stages we were `all over the park at times` and to come back from a goal down was a welcome bonus. But when David Weatherston came on his speed made all the difference.
"I suppose we could say we `dogged it` out there today but both strikes from `Magic` Johnston and Rocco Quinn were that bit special."
Ian McCall however decribed the result as `one hell of a blow` and said he felt sorry for the Thistle punters. And he insisted, "We were in total control for most of the game but individual errors at the end cost us dearly.
"Queens had only two real shots at goal and netted twice, but we have only ourselves to blame as we should have had all three points tied up long before the end. It`s been a horrendous day and I`m gutted."
Queens Man of the Match - Craig Reid - An outstanding display by the Queens defender.