Stewart Downing's 25-yard effort midway through the second half was enough to give the Reds the edge going into the second leg match at Anfield next Thursday.
Gomel dominated the majority of the game and the Merseysiders were lucky to leave with the win after the hosts missed plenty of chances.
Liverpool's back four looked rocky throughout and it was poor finishing rather than the performance of goalkeeper Brad Jones which cost the home side.
Gomel, who are halfway through their domestic season, were full of energy and pressed Liverpool high up the pitch, giving the visitors little time to dwell on the ball.
Rodgers promised he would take the competition seriously and picked a strong side although Daniel Agger and Andy Carroll - both of whom have been linked with moves away from Anfield - stayed in the north-west.
Steven Gerrard was captain, while Jamie Carragher made his 700th Liverpool appearance and was partnered in the heart of the defence by Martin Skrtel.
Joe Cole, who spent last season on loan at Lille, also started - while summer signing Fabio Borini led the line in a 4-3-3 formation.
However former Chelsea man Cole lasted just 23 minutes as he hobbled off with what appeared to be a hamstring injury, with 17-year-old Raheem Sterling his replacement.
Both teams began slowly although the visitors showed they were already buying into the Rodgers approach, with the vast majority of their football played on the floor.
Sadly for the Merseysiders it failed to yield opportunities and it was Gomel who created the first chance 25 minutes in.
Dmitri Platonov skipped past Gerrard and surged to the byline before delivering an accurate cross to Igor Voronkov, who headed over from close range.
Five minutes before the break and central defender Martin Skrtel was first outmuscled and then outpaced by striker Aleksandr Alumona, who shot straight at Jones - with the Australian deflecting the effort wide.
The danger was far from over and a short corner from Gomel caught the visitors out and Platonov should have tested Jones with a free header from 10 yards out, but mistimed and looped his effort over.
The hosts took just a minute to create an opportunity in the second half as once again the backline fell asleep as Platonov was played onside by Carragher and raced away down the left. The forward rounded Jones but his touch was too heavy as the ball ran away for a goal-kick.
Liverpool had another let off and it was Platonov who again went close after 65 minutes. Jay Spearing gave the ball away cheaply and Sergey Kozeka hit a rasping shot which cannoned onto the post with Jones well beaten. The rebound fell to Alumona who completely mishit his effort into the path of Platonov who, with the goal gaping, somehow shot wide.
Gomel were made to pay for their incompetence in front of goal as just two minutes later - following the introduction of Lucas for Jordan Henderson - the visitors were in front.
Downing cut inside from the right and unopposed from 25 yards fired past Vladimir Bushma into the left hand corner.
The goal deflated the home side, with Liverpool keeping possession well for the remainder of the game.
The visitors could have doubled their lead late on as Borini, who lacked service all night, burst through. However Bushma forced the Italian wide and parried the striker's effort away from danger.
However Liverpool left with a win and a clean sheet as they seek to make sure of an extended run in this season's Europa League.