Morton reached their first major semi-final in 35 years with a hard-fought 2-1 Betfred Cup win over fellow Championship side Dundee United at Cappielow.
Morton attacker Jai Quitongo blasted the home side into the lead in the 24th minute before defender Thomas O’Ware added a second just after the half-hour mark with a header.
Nick van der Velden reduced the deficit in the 55th minute with a drive but Jim Duffy’s side held out for a landmark quarter-final victory.
Morton were beaten 2-1 by Rangers in the 1981 Scottish Cup semi-final - their last League Cup semi-final was against Alex Ferguson’s Aberdeen in 1979 - and the Greenock club will want to go one better this year.
Duffy’s side go into the draw with Rangers, while in the other quarter-final ties Celtic are to play Alloa on Wednesday and Aberdeen face St Johnstone on Thursday.
The home side went into the game with some much-needed confidence after ending a run of five games without a win - and three straight defeats - with a 2-1 home win over Dunfermline.
United looked to get back on track having lost 3-1 to Falkirk, but neither they nor Morton could settle in the disjointed early stages.
In the 16th minute Van der Velden played in United forward Cammy Smith with a clever pass but his angled shot missed the target.
However, when Morton got their first real chance they capitalised.
Quitongo raced clear of the United defence after latching on to a Ross Forbes head flick, and despite being chased by defender Frank van der Struijk he kept his composure to lash the ball high past goalkeeper Cammy Bell.
The Greenock side’s second goal soon followed when O’Ware leapt high inside the United penalty box to bullet in a Forbes corner.
Suddenly, United were facing a cup exit.
The visitors at last realised their predicament and began to play with purpose and Morton goalkeeper Andrew McNeil made a decent save from Scott Fraser’s drive just before the break, the home side defending the corner.
Quitongo threatened twice on the counter after the teams changed ends, but Van der Velden changed the complexion of the tie when he fired low past McNeil after a Fraser header left him with only the goalkeeper to beat.
A rejuvenated United went flat out for the equaliser and in the 72nd minute McNeil pushed a Blair Spittal drive past the post, before Forbes missed the target at the other end with a long-range effort but only by inches.
Morton were on the ropes towards the end as the Taysiders flooded forward, but the hosts battled through four added minutes for an historic win.