The home team had the first real chance of the game in the 14th minute.
Russell Latapy's cut back was poorly cleared by the Clyde defence and the ball fell to Scott McKenzie, whose low driven shot was turned out for a corner.
From that flag-kick Falkirk gaffer John Hughes' header was cleared off the line by Gary Bollan.
Fourteen minutes later, a throughball from Darryl Duffy fell to Daniel McBreen, who was sandwiched between two defenders but managed to stop and shoot - and it took a fine stop by Clyde keeper Bryn Halliwell to save the day.
The visitors came close in the 34th minute. A Simon Mensing header from a corner produced a fine instinctive save by Falkirk keeper Allan Ferguson.
Two minutes later the home side were again close to breaking the deadlock when Duffy tried to chip the keeper, but Halliwell was not about to be fooled.
After the break, a chance fell to Latapy directly from a corner, ably assisted by the driving wind, but it was again beaten away by Halliwell.
Other chances fell to McBreen, who failed to convert with only the keeper to beat, and Latapy, with an opportunist shot from outside the box.
But in the end a draw was a fair result.