Celtic were held to a 1-1 draw by Lithuanian champions Suduva on Thursday but remain on course to reach the Europa League group stage ahead of the home leg.
Having been denied qualification for the Champions League for the first time under Rodgers, Celtic wasted little time in their Europa League play-off, taking the lead after just two minutes when Michael Johnston shook off his marker with a neat turn and whipped in a cross that was met by a powerful Olivier Ntcham header into the bottom right corner of the net.
But the visitors' defence was caught napping 10 minutes later when Vaidas Slavickas' looping free-kick from the right found Ovidijus Verbickas unmarked in the box and the midfielder nodded the ball down beyond Craig Gordon's reach.
Johnston squandered a gilt-edged chance to put Celtic back in front, heading straight into the arms of goalkeeper Ivan Kardum from 10 yards and Moussa Dembele's curling 30-yard free-kick was also saved on the cusp of half-time as the home crowd sensed the potential for an upset.
Dembele hammered a shot into the side-netting early in the second half after latching onto Ntcham's clever throughball, before an acrobatic save from Gordon kept the scores level as the Celtic keeper leapt to push Rigino Cicilia's rising shot over the crossbar.
Kieran Tierney lashed an angled shot narrowly over the crossbar from 25 yards but Suduva continued to press on the counter, Gordon saving smartly when substitute Julius Kasparavicius fired in a shot from close range.
Rodgers sent on Leigh Griffiths to try and force a late winner that proved elusive despite Celtic's late dominance, but they will fancy their chances of a more comfortable evening at Celtic Park in the second leg.