David Hopkirk cancelled out Reece Cole's long-range effort with around 20 minutes remaining to give the Warriors hope of a Cup upset and a place in the last four.
However, the Jags immediately climbed back in front through substitute Gary Harkins.
Before the visitors came good in the end through further strikes from Craig Slater and Kenny Miller.
There may have been two divisions between the sides but there was no evidence of that in the opening stages.
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The Warriors started brightly, pressing and harrying and given the visitors no time on the ball, with Thistle struggling to get out.
That said, it was Ian McCall's men who thought they had the lead on 13 minutes but they were denied by a stunning save from Graeme Smith.
Penrice surged down the left, planted the ball on Mansell's head but somehow Smith reacted instinctively to tip over the header from point blank range.
Thistle grew more into the game as the half wore on and Joe Cardle should have done better than skew his volley wide at the back post, before Kenny Miller headed over from close range.
But the Jags did find a breakthrough and it was a fine stike from Brentford loanee Cole who fired in a right foot swerving effort into the bottom corner from distance.
The Warriors had not been seen much in an attacking sense and an Andy Munro header which drifted wide was as closest as they came in the opening hour.
At the other end, Thistle looked threatening. Slater should have done better than shoot tamely at Smith before Wiliamson cut inside and fired in a left foot effort which curled wide.
Stenny stunned their Championship opponents when they levelled on 68 minutes thanks to Hopkirk.
Dykes did well to hold the ball up after a long ball into the box, he played it down to the attacker who drilled a low shot beyond Fox.
The hosts were level for only a couple of minutes as substitute Harkins bundled home from close range after receiving a knock down from O'Ware following Cole's corner on 70 minutes.
And the Jags sealed the match nine minutes later when Slater, who had previously been denied just minutes before, slid the ball under Smith from Cardle's pass on 79 minutes.
Miller put gloss on the scoreline as he found space inside the box and rifled the ball into the top corner on 86 minutes.