Lyndon Dykes opened the scoring before Conor Sammon netted a dramatic equaliser for the Bairns to make it 1-1 at full-time.
Livi scored all their spot-kicks and Sarkic saved from Michael Tidser and Lewis Toshney to hand Gary Holt a winning return to the Falkirk Stadium.
Livingston started brightly in a Group G encounter, with confusion in the box as Falkirk defender Mark Durnan headed against his own post.
But it was the Bairns who fashioned the first effort on goal when Charlie Telfer's curling effort from the edge of the box was saved by Sarkic. New skipper Gregor Buchanan headed over the resulting corner.
Chances were at a premium in the match but Holt's men opened the scoring in the 36th minute.
Dykes played a good one-two with Scott Robinson before running in on goal, his shot clipping Bell on the way in as it looped over the helpless goalkeeper for his first goal for the club.
Livi, buoyed by the goal, could have added a second minutes later, as Dykes cut in on his right and fired a thunderous effort towards goal, but Bell saved.
Falkirk brought on Denny Johnstone in place of Aidan Connolly and almost grabbed an equaliser minutes after coming on when he prodded his shot wide at the far post.
But they secured a draw in dramatic fashion when Sammon fired in from close range before Bell ensured the match would go to penalties with a great save from substitute Aymen Souda.
Sarkic then denied the Bairns in the shoot-out.