Kilmarnock won the match 1–0, the only goal coming from Paul Wright in the 20th minute of the first half.
In the fourth round the club was drawn away to Clyde and produced a narrow 1–0 victory to progress to the quarter-finals.
In the semi-finals, Kilmarnock was drawn against Dundee United with the match played at the neutral venue of Easter Road in Edinburgh.
The clubs played out a 1–1 draw with the replay a week later resulting in a 2–1 victory for Falkirk to enter the next round.
Falkirk emerged 1–0 winners in the replay at Ibrox, progressing to their first in forty years since defeating Kilmarnock in a repeat of the fixture in the 1957 competition.