The Bairns played the game "under duress" after the Scottish Professional Football League board rejected their postponement plea following a spate of infections.
Nine players came out of self-isolation on Saturday morning, and they looked to be heading for the points after Aidan Nesbitt's second-half strike put them ahead.
Nesbitt struck midway through the second period, converting after a goalmouth flurry.
But Rory McAllister levelled 13 minutes from time for the hosts to earn a point and disappoint Falkirk
Montrose started the season well with a 3-0 win at Airdrie.
They went ahead in the 10th minute as Terry Masson latched on to Lewis Milne's impressive ball and fired home.
The favour was returned five minutes after the break as Masson fed Milne, who proceeded to beat his man and arrow a low shot into the bottom corner.
Graham Webster's penalty 16 minutes from time wrapped up the points.
Dumbarton also won 3-0 with a sound beating of Clyde.
Conrad Balatoni put through his own net as the visitors went ahead early on and Ross MacLean's second before half-time strengthened their position.
Conner Duthie then sealed the game 10 minutes from time.
Goals from Russell McLean and Niah Payne gave Peterhead a 2-0 win over Alloa, while East Fife and Queen's Park drew 1-1.