A pair of crucial saves from goalkeeper Michael McGovern kept Falkirk on the promotion pace.
The Bairns' number one pushed away Peter MacDonald's 49th minute penalty then tipped a Michael Tidser drive on to the post in the final act of the game.
With Raith holding Ross County and Dundee drawing 2-2, the Bairns are no worse off in the race to the SPL.
Falkirk fielded Jordan White and Jay Fulton in an unorthodox forward line, and later injected Willie Gibson and Craig Sibbald into the attack - but still couldn't find the net.
Colin Stewart in the opposition goal was always on guard but never truly tested by the visitors to the Clyde coast.
MacDonald had the best chance to break the deadlock after Tam Scobbie was judged to have handled inside the box.
The ball seemed to strike the left-back on the upper body and Steven Pressley claimed he was outside the penalty area when Fouad Bachirou's stirke was charged down, but ref Bobby Madden disagreed and pointed to the spot.
In the first half White had volleyed wide for Falkirk from close range and Stewart Murdoch had a second half header fly past the upright.
In the final minute Michael Tidser looked to have won it with the last kick of the game, but McGovern tipped the ball onto the post and regathered the rebound.
Elsewhere Stenhousemuir lost 2-1 at home to Dumbarton and had Kevin McKinlay, Ross McMillan and Davie Irons all sent off. The manager was banished to the stands after the dismissal of McMillan early in the game.
With nine men at half-time - and the enforced substitution of Brown Ferguson - the Warriors battled to claw a goal back in the second half.
East Stirlingshire conceded three goals on the road at Clyde.
Dunipace Juniors defeated Vale of Leven 2-1 at Westfield.
For more on these matches, see this week’s Falkirk Herald.