Though the Reds were under pressure for much of the contest, Stewart Downing grabbed the game’s only goal when he found the bottom corner with a powerful shot from the edge of the box in the 67th minute.
Liverpool was able to see the remainder of the first leg of the UEFA Europa League third-round qualifier out and ensure the new manager marked his first competitive game in charge with a win.
There were plenty of other plus points for the Reds, as Carragher chalked up his 700th appearance for the club while Lucas Leiva was introduced into the action as a second-half substitute as he continues on the road to full fitness.
Rodgers named a strong line-up for the clash in Belarus — though it was the selection of Carragher that grabbed the headlines as he notched up his 700th run-out for the Reds.
In recognition of the feat, the veteran defender was named as the team’s captain for the night, despite the inclusion of Steven Gerrard in the starting lineup.
Meanwhile, Fabio Borini was handed his competitive debut for the club as he led the attack, while Glen Johnson reached the 100-game mark for Liverpool.
Gomel created the first chance of the evening with two minutes on the clock as Igor Voronkov strode forward from deep and let fly with a dipping effort that grazed the roof of Brad Jones‘ net.
The hosts certainly made an energetic start to the contest, with home fans again rising to their feet when Artur Liavitski cannoned a strike against the legs of Martin Skrtel in a dangerous position on the edge of the box.
But they were almost undone when Borini hooked Downing’s corner wide of the near post soon after, before Johnson headed a Gerrard free-kick over the top.
Liverpool was then forced into making a change in the 22nd minute when Joe Cole pulled up. Rodgers opted to call on Raheem Sterling to replace the winger.
And the visitors were fortunate not to find themselves a goal behind moments later as Dzmitry Platonov skipped beyond the challenge of Gerrard and delivered an inviting centre, which Voronkov nodded into orbit when well positioned six yards out.
Gomel went closer still when Alexander Alumona turned sharply and drilled over after collecting the ball inside a crowded area from a corner.
Alumona’s threat was in the ascendancy and after he’d stung the palms of Jones with an angled strike, the diminutive forward sent a 20-yard whack wide of the upright.
With six minutes of the half remaining, Alumona again caused panic as he this time held off the attentions of Skrtel as the pair both chased a long ball forward, before seeing his attempts to slide the ball beyond Jones from close range thwarted by the legs of the custodian.
Next, only a last-ditch block from Carragher — the likes of which he’s become famed for over the last 700 games — prevented Alumona tapping into an empty net from the unmarked Platonov’s cut-back.
From the resulting corner, Mikalai Kasheuski lofted over to again have the Gomel fans holding their heads in their hands as the Belarusian side went into half-time delighted with the opening 45 minutes.
Rodgers will no doubt have demanded better from his men during the break, though the boss was required to make another change, with Martin Kelly taking the place of Johnson.
And Gomel attempted to pick up where they left off as a long ball forward allowed Platonov to spring Liverpool’s offside trap and bear down on Jones’ goal. The forward managed to knock it beyond the onrushing goalkeeper, though his touch was too heavy and took the ball out for a goal kick.
The Reds were next to threaten, however, with Borini seeing a low drive on the spin repelled by Vladzimir Bushma before Tomasz Nowak flicked a Gerrard set-piece inches wide of his own goal.
But only an extraordinary moment of play kept the scoreline goalless in the 65th minute. Siarhei Kazeka took aim from distance and saw his strike cannon against the post. The ball broke for Alumona, who inadvertently played it into the unmarked Platonov with a mis-kick. Thankfully for Liverpool, though, he somehow contrived to steer wide when a goal appeared inevitable.
And Gomel was made to pay for its profligacy when Downing fired Liverpool into the lead two minutes later.
Collecting the ball outside the home side’s area, the winger struck an unstoppable effort into the bottom corner of the net to put the Reds ahead.
One goal was enough to secure victory for Liverpool in Belarus and help them take a big step towards the Europa League proper ahead of next week’s second-leg meeting at Anfield.
Meanwhile, for Rodgers, he’ll be hoping the victory at the Central Stadion is just the first of many during his tenure.