Sibbald nodded in Luke Leahy’s driven cross in the 35th minute at an extremely windy Palmerston Park on Friday night to prevail in the all Scottish Championship clash.
With tricky conditions, it proved difficult fot the players to control the ball but the contest livened up with a long-range volley from Falkirk striker Rory Loy and it proved the stimulus for the visitors to take the lead.
The goal came from Leahy’s throw-in, with the left-back powering in a powerful volleyed cross which swerved beyond the home defence.
Sibbald ran on to it and got the slightest of touches to steer it away from Clark and into the far corner.
Falkirk goalkeeper Jamie MacDonald kept his side in front at half-time when he blocked Gavin Reilly’s shot from a tight angle and the Bairns held out well in the second period.
The closest the home side came was when substitute Dan Carmichael beat Leahy and put in a dangerous ball which was half-cleared to Russell, but the forward’s shot bounced into the ground and up into MacDonald’s arms.
The only downside to Falkirk’s cup exploits was two injury blows in the final 10 minutes with Loy going off with an ankle knock and Leahy picking up a head injury after being kicked by team-mate Grant with the left-back attempting a headed clearance.
There were seven minutes of injury-time after Leahy was carried off on a stretcher and Falkirk stood up to an aerial barrage to make the last four for the second time in three seasons.