A 1-1 draw at home to bottom club Morton, in which they missed a penalty, looked insufficient with closest rivals Dundee and Hamilton both leading late on.
But neither could hold onto their leads and the top three clubs all had to be content to share the spoils.
Falkirk should have taken the lead six minutes from half-time, but Rory Loy missed from the spot for the second successive week, with keeper Derek Gaston also blocking his follow-up.
Dougie Imrie combined well with Garry O'Connor to put the Greenock side ahead on the stroke of half-time, but Falkirk stretched their unbeaten league record when Mark Millar equalised after a second penalty award midway through the second half.
A goal - and another from the penalty spot - by Ryan Conroy after 69 minutes looked enough to give Dundee victory against Alloa Athletic on former manager Barry Smith's return to Dens Park.
But Smith was to finish the happier boss after Ben Gordon struck deep into injury-time to give Alloa a draw.
Dundee should have killed the game off before then with Martin Boyle guilty of some tame finishing, but Alloa also had their moments and Dan Twardzik had decent saves in the first half from Stephen Simmons and Ross McCord.
Hamilton's chance to go top was ruined by some poor defending as they shared a 3-3 draw with Dumbarton at New Douglas Park. James Keatings put the home side ahead early on, but Jordan Kirkpatrick equalised before half-time when he was left unattended at the back post.
Hamilton then fell behind in the second half when Andy Graham applied the final touch to a free kick.
Keatings equalised and Jason Scotland put them in front with a finish from close range, but the home celebrations had barely died down when Chris Kane headed an equaliser with just a couple of minutes left.
Queen of the South moved into the top four with a 2-1 win at Livingston. The home side took the lead when Mark Burchill was adjudged to have been fouled by Chris Mitchell as the striker leapt to meet a Mike Mampaya cross. Marc McNulty dispatched the penalty.
Mark Durnan equalised after the break when he headed in Michael Paton's deep cross and substitute Derek Lyle pounced on a rebound to score the winning goal for Queen of the South.
At Central Park, Kane Hemmings scored the only goal of the Fife derby as Cowdenbeath beat Raith Rovers 1-0.
The game did not come alive until the second half, but Hemmings looked the most likely player to break the deadlock.
He tested Ross Laidlaw in the only first-half effort worthy of note and his pace troubled Raith all afternoon.
The only goal came after 70 minutes when Hemmings scored with a diving header.
It gave Cowdenbeath their first home win in the league since October 19 and ended a six-match losing sequence for them at Central Park.