Luis Díaz had a clinical finish into the roof of the net in the 63rd minute to put the visitors ahead after receiving a low cross from midfielder Ryan Gravenberch, who was given his first start for the Reds since his summer transfer from Bayern Munich.
Mohamed Salah came off the bench and completed the second-half comeback with a goal in the 88th minute.
Florian Flecker beat backup goalkeeper Caoimhin Kelleher off a corner kick in the 14th minute for the opener against Liverpool, a three-time winner of the second-tier European competition.
The visitors equalized when Darwin Núñez scored on a penalty kick 11 minutes into the second half after Philipp Ziereis fouled Díaz.
Liverpool manager Jürgen Klopp made 11 changes in his starting lineup from the 3-1 comeback victory over Wolverhampton on Saturday in the English Premier League.
It was a return to the second-tier competition for the Reds for the first time since losing 3-1 to Sevilla in the 2016 final in Basel in Klopp’s first season at Anfield.
In another Group E game, Union Saint-Gilloise and Toulouse drew 1-1.
Romelu Lukaku scored the second-half winner for last year’s runner up Roma in Group G in a 2-1 victory over Sheriff in Tiraspol. It was Lukaku’s second goal since joining from Chelsea.
Earlier, substitute midfielder Leandro Paredes netted from a deflected free kick in first-half stoppage time to give the visitors a 1-0 lead before Cristian Tovar evened it in the second half.
Roma, coached by Jose Mourinho, is trying to reach their third straight European final.
Also in Group G, Slavia Prague took home three points from Geneva after beating Servette 2-0.
Ludovic Blas scored the first goal in the competition this season in the second minute to help Rennes beat Maccabi Haifa 3-0 in Group F. In the group’s other game, Panathinaikos defeated Villarreal, the 2021 Europa League champion, 2-0.
In Group H, Florian Wirtz, Aminde Adli, Victor Boniface and substitute Jonas Hofmann had a goal each for Leverkusen to rout Sweden’s newcomer Häcken 4-0.
EUROPA CONFERENCE LEAGUE
Albanian forward Ernest Muçi netted twice in the second half in Warsaw as Legia beat Aston Villa 3-2 to spoil the English team’s return to the the group stage of a European competition for the first time since the 2008-09 season.
Villa, the 1982 European Cup winners, had to come from a goal down twice in the first half in the Group E game.
Villa forward Jhon Duran headed in an equalizer just three minutes after Pawel Wszolek put the hosts ahead in the third minute. Muçi made it 2-1 in the 26th with his first before defender Lucas Digne made it 2-2 later on.
In the other group’s game, Zrinjski made a victorious debut for a club from Bosnia in Europe with a 4-3 win over Alkmaar.
In Group F, last year’s runner-up Fiorentina started their campaign with a 2-2 draw at Genk in Belgium. Defender Luca Ranieri headed home twice for his first two goals for the Italian team.
In Group G, Eintracht Frankfurt opened with a 2-1 over Aberdeen.