The breakthrough came on 26 when Chris Burke slipped a ball through to Nacho Novo from the right flank for the striker to stab home from the edge of the six-yard box.
Thistle could have equalised had Simon Donnelly’s smart low shot not been blocked – instead they found themselves trailing by a second goal six minutes from the interval.
A deep cross from Novo found Kris Boyd at the back post who nodded to Burke, allowing the Ibrox winger to loop a header over the goalkeeper.
Partick were thrown a lifeline just four minutes after the restart when referee Iain Brines pointed to the spot.
Donnelly stepped up for the penalty but he blasted wide.
Rangers will now face St Johnstone in the last four.