In an all Championship replay at Cappielow, Morton – who are without a manager following the departure of Gus MacPherson at the weekend – took the lead through Robbie Muirhead after good work from Reece Lyon on 42 minutes.
However, Billy Dodds’ outfit responded and equalised two minutes into the second half thanks to Irish defender Danny Devine.
That was the end of the goalscoring and after the first eight penalties were scored, Devine missed his, allowing Gary Oliver to score the winning spot-kick and secure a 5-4 victory.
The Greenock outfit will now travel to Fir Park on the weekend of January 22 in a repeat tie of last year.
Former Hibs midfielder Fraser Fyvie opened the scoring on 38 minutes before two late second-half goals in quick succession for Mitch Megginson sealed the win for Paul Hartley’s men.
In League One, Montrose moved into second place with a 2-1 victory at home to Airdrieonians.
The hosts went ahead on 20 minutes thanks to a goal from forward Craig Johnston.
Kerr McInroy responded for the visitors, but Stewart Petrie’s men found the winner via midfielder Graham Webster.
Montrose are now three points behind leaders Cove Rangers on 28 points, while Airdrieonians remain in the play-off places, two points further behind in fourth place.
Meanwhile, in League Two, Kelty Hearts’ match away at Elgin City was called off less than 90 minutes before kick off due to a waterlogged pitch stemming from Storm Barra.
Kelty manager Kevin Thomson, however, was left less than impressed with the decision to postpone the match so late in the day.
The former Rangers and Hibs midfielder took to Twitter and wrote:
He tweeted: "Shambles asking part time players, take days off work, 6 days notice to travel through a yellow warning, 4 and half hours on bus to get called off!!
"Some common sense would be nice to be part of the thinking behind scheduling."