The Ibrox men will now face St Johnstone at Hampden in next Sunday's semi-final.
In a lively start to the game, Thistle were first to look threatening when Mark Roberts released Damon Gray after six minutes, but the striker's shot was parried by Gers keeper Allan McGregor and the loose ball fell to Kevin Twaddle who shot narrowly wide.
Rangers responded two minutes later will a well-struck free-kick from Kevin Thomson which was saved brilliantly by Jags keeper Jonathan Tuffey.
As the action flowed to the other end, an excellent move involving Roberts and Gary Harkins resulted in an opening for Simon Donnelly but the one-time Celtic midfielder sent his shot wide of goal.
Thistle were looking increasingly comfortable as the first half progressed but they were stunned in the 27th minute when Nacho Novo fired Rangers into the lead, the diminutive Spaniard stretching out to meet a low cross from Chris Burke and deflecting the ball beyond Tuffey.
The outcome of the match hinged on two incidents just before the interval.
Firstly, Donnelly saw what appeared to be a goal-bound effort saved by McGregor and then, seconds later, Rangers mounted an attack which culminated in Burke heading home after Steven Smith's cross had been nodded back across goal by Kris Boyd.
The home side were given a great chance to stage a comeback early in the second half when referee Ian Brines awarded a penalty after David Rowson had been tripped by Smith, but Donnelly sent the spot-kick wide of McGregor's right-hand post.
Donnelly then saw his close-range shot clip the crossbar as it became clear that this just wasn't going to be Thistle's day.
Having seen off Thistle's efforts to fight their way back into the game, Rangers regained control towards the end and they should have increased their lead two minutes from time when Willie Kinniburgh tripped Burke in the box.
Referee Brines had no hesitation in awarding the penalty but Rangers substitute Daniel Cousin became the second player to miss from the spot as Tuffey saved the Frenchman's weakly-struck effort.