Genk 1-4 Liverpool: Champions League Group E – live!
Alex Oxlade-Chamberlain scored twice, including a stunning outside-of-the-boot finish, but said Champions League holders Liverpool must "improve" despite hammering Genk in Belgium.
The Reds' 100% start to the Premier League season ended with a draw at Manchester United on Sunday, but they returned to winning ways by showing their class in the Group E match.
England international Oxlade-Chamberlain, who missed almost all of last season with a knee injury, drilled in a low shot in the second minute and brilliantly converted a second with a cool, first-time finish from Roberto Firmino's pass.
Sadio Mane dinked in the third from Mohamed Salah's through ball and the Egyptian turned past a defender and rolled home the fourth late on.
Liverpool face Tottenham in the league on Sunday (16:30 GMT) and Oxlade-Chamberlain said: "We have gained confidence from the whole season, not just this game and the last 20 minutes against Manchester United.
"We saw that we have a lot to improve on and it was the same again tonight, but we're getting results, which is good.
"The manager will tell us we have a lot of points to improve because we've got to get them right if we're to get another good result at the weekend."
Belgium champions Genk had responded well to falling behind and were unfortunate not to equalise - striker Mbwana Samatta dragged a shot wide and had a towering header ruled out for a tight offside call.
Striker Paul Onuachu had a low shot palmed out by Alisson, and the Brazilian goalkeeper was eventually beaten with two minutes left through a Stephen Odey consolation.
Liverpool remain second in the group, a point behind leaders Napoli who came from behind to beat Red Bull Salzburg.
Liverpool have consistently produced not only in the Premier League but also in Europe's elite club competition over the past two seasons, lifting the trophy for the sixth time in June after losing in the 2018 final.
Despite impressive knockout away victories during that period, including 5-0 and 4-1 in Porto, 3-1 at Bayern Munich and 2-1 against Manchester City, a quirky statistic had seen Liverpool lose their previous four games on their travels in the group stages.
This included a reverse against Napoli in the opening game of this season's competition but the Reds have responded by collecting back-to-back wins to consolidate their position in the group.
They kept Gaetan Coucke busy - Mane's effort from close range, Robertson's drive and Firmino's dipping strike all kept out by the 21-year-old Genk goalkeeper.
Salah, who missed the Manchester United game because of an ankle injury, fired a free-kick narrowly over and although he and Mane got on the scoresheet, boss Jurgen Klopp may be concerned by his side having kept only three clean sheets in 15 matches in all competitions this season.
They were, at times, rather easily opened up at the back by Genk, as two long balls set Samatta and Onuachu in on goal, but neither player was able to find the net.
The result means Genk have now played 15 matches in the competition without winning - more than any other side.
Oxlade-Chamberlain's 2018-19 season was a write-off, a serious knee injury restricting him to just two games all campaign.
A player well-liked by Klopp and the Liverpool supporters, the ex-Arsenal midfielder gave his side control with two well-taken strikes.
He came close to opening his account for the season at Old Trafford but managed to rasp home early against Genk, collecting Fabinho's pass, spinning and drilling in a low finish from 25 yards.
His second was sublime, again from outside the area, from a lay-off by Firmino - the Brazilian taking his tally to 11 assists, in addition to scoring 14 goals, in the Champions League since the start of the 2017-18 season.
With regulars Jordan Henderson and Georginio Wijnaldum left out of the starting line-up, Oxlade-Chamberlain and the industrious Naby Keita highlighted Liverpool's squad depth.
The Guinea midfielder is another who has been beset with injuries in his time at Anfield but worked tirelessly all evening, making more passes (118) than any other player, winning the ball back 12 times and contributing three tackles.
Klopp ready for 'refreshed' Tottenham
Liverpool manager Jurgen Klopp told BT Sport: "There were good moments. The start was brilliant, we were kind of fluent and asked them a lot of questions but then we started - I'm not sure why - to lose patience and some easy balls. But all four goals were brilliant and we had chances to score more. Job done.
"In the years before we would have drawn it, maybe even lost, but all's OK.
"We've just had two away games that were both difficult for different reasons. I don't know where we are now but I know where we need to be on Sunday.
"It's a refreshed Tottenham. Of course the second they meet us they're back on track. But we're a different team to Red Star Belgrade, just like Tottenham are different to Genk.
"Oxlade-Chamberlain's goals were brilliant, and very important. We're really pleased for him, it was a big step."
