Falkirk edged past local rivals Stenhousemuir 1-0 to reach the fifth round.
Kieran Duffie's ninth minute goal separated the sides as the First Division club avoided an upset.
Duffie collected Stewart Murdoch's pass and rifled a low left-footed drive past Callum Reidford from 12 yards.
The striker was denied a second as his well-struck effort from the edge of the area was well held by Reidford in the 23rd minute.
Sean Dickson went closest to forcing a first-half equaliser but his header was straight at Michael McGovern.
Conor McGrandles and Thomas Grant were both denied by Reidford in the second period, while Darren Smith's header flashed just wide for the home side with 11 minutes left.
Falkirk successfully saw out the remaining time to secure their safe passage to the next round.