The home side failed to impose themselves until the final quarter but, other than an excellent free-kick from debutant Ki Sung Yeung, they could not trouble Robert Olejnik.
Falling further behind in the title race further called into question the decision to sell the influential Barry Robson to Middlesbrough and Gary Caldwell, last season's Scottish Football Writers' player of the year, to Wigan.
Late withdrawals from the injured Danny Fox and ill Glenn Loovens meant Tony Mowbray was without six players who featured in the Old Firm draw on January 3 with Landry NGuemo on African Nations Cup duty and Scott McDonald recovering from hernia surgery.
With Jos Hooiveld, a midweek signing from AIK Stockholm, short of fitness, Mowbray paired central defenders Stephen McManus and O'Dea, who was making his first appearance since October, which came during a loan spell at Reading.
Lee Naylor came in at left-back and Zheng Zhi returned to midfield while Falkirk handed debuts to former Leeds striker Enoch Showunmi and on-loan Ipswich midfielder Colin Healy.
Samaras headed wide from a Marc-Antoine Fotune cross inside 30 seconds but Celtic were soon on the backfoot and Andreas Hinkel cleared Pedro Pele's header off the line following a Ryan Flynn corner.
Ki was playing in the deep position of a midfield diamond spearheaded by Aiden McGeady, and Celtic's best early move started and finished with the pair. Zhi burst past the Falkirk midfield after Naylor had helped on Ki's pass and fed McGeady, who shot just wide from 20 yards.
Fortune headed Ki's deep free-kick just beyond the far post but Celtic soon fell behind. Finnigan knocked the ball past O'Dea after a Showunmi head-on but the Irishman seemed in control until the striker brushed him off and sent Artur Boruc the wrong way from 16 yards.
Boruc rescued Celtic in the 28th minute, parrying Flynn's measured shot from 18 yards after Finnigan had laid off a Marc Twaddle cross.
Marc Crosas shot just over from similar range after the Falkirk defence struggled to clear a McGeady cross but Celtic soon suffered another blow when McManus was replaced by 18-year-old Josh Thompson.
The hosts were struggling to break down a hard-working Falkirk side but Samaras worked some room away from Pele on the left channel and fired inside the far post to level the match after 40 minutes.
Naylor curled a 20-yard free-kick just over and Zhi headed Hinkel's cross wide from close range after the restart but the visitors were enjoying an equal share of possession and Vitor Lima twice came close from long range.
Ki almost made himself an instant Celtic hero in the 66th minute with a brilliant free-kick, however Olejnik flew to his left to push the 30-yard effort away from his top corner.
Winger Niall McGinn replaced Crosas as Mowbray reverted to a more familiar midfield shape with McGeady moving wide left, and Celtic stepped up the pressure.
But Lima blocked well when McGeady worked himself into a shooting position after a short corner and Fortune miscued Ki's long free-kick.
Paddy McCourt came on for Hinkel but and quickly beat two men and sent over an inviting cross, but there were no takers, and Zhi's powerful cross soon flashed wide off the head of Samaras.
Boruc was penalised for carrying the ball out the box to the disgust of the home fans but sub Jack Compton was a yard too high with the resulting free-kick.
The Celtic support soon vented their ire against referee Alan Muir when he ignored penalty appeals after Fortune fell hopefully inside the box under the attentions of McLean.
McCourt fired a 20-yard shot just wide in the 89th minute with a slight deflection and Falkirk held on for their second draw against Celtic this season.