Mohamed Salah, who scored twice, grabbed the 69th-minute winner for Liverpool after Salzburg, managed by American Jesse Marsch, had stunned Anfield with three goals either side of halftime to make it 3-3.
Champions League holders Liverpool came out on top of a 4-3 thriller with Salzburg on Wednesday after throwing away a three-goal lead at Anfield before Mohamed Salah's winner, while Luis Suarez fired Barcelona to a comeback victory over Inter Milan.
Salah won the Group E match for Liverpool with his second of the night in the 69th minute, denying Salzburg a point after teenage sensation Erling Braut Haaland had stunned the Reds when he completed an incredible fightback with his 18th goal of the season.
The hosts looked to be cruising when Egyptian Salah had added to goals by Sadio Mane and Andy Robertson with a neat finish in the 36th minute, but Hee-Chan Hwang's brilliant strike shortly before the break gave Salzburg hope.
Two goals in four second-half minutes from Takumi Minamino and 19-year-old substitute Haaland, who had only been on the pitch for four minutes when he evaded a dozing Liverpool defence to tap in Minamino's low cross, left the home crowd stunned.
However, Salah spoiled their evening with his sixth goal of the season in all competitions, putting Liverpool level with Salzburg on three points, one behind Napoli after their disappointing goalless draw at Genk.
Suarez saved Barcelona's blushes at the Camp Nou after Lautaro Martinez had scored early for Inter, who then failed to capitalise on their first-half dominance in the Group F clash and lost 2-1.
Marc-Andre ter Stegen had to be at his best in the 37th minute to get down low and keep out Martinez's powerful header, while Stefano Sensi also rued missing the target when well placed shortly afterwards.
However Barca moved up a gear in the second half and were deservedly level when Suarez crashed home a superb shot just before the hour mark, and the Uruguayan was at his sharpest to win the match with six minutes remaining.
Lionel Messi, a surprise inclusion from the start, suddenly warmed up and skipped past two challenges before laying on Suarez, whose perfect first touch allowed him to sink Inter and draw Barca level on four points with group leaders Borussia Dortmund.
Flamboyant Ajax shine again
The Bundesliga outfit came out on top at Slavia Prague thanks to a pair of counter-attacking strikes from Achraf Hakimi, whose clinical finishing led Lucien Favre's side to a 2-0 win in the Czech Republic.
The Morocco international burst through onto Julian Brandt's pass in the 35th minute and charged towards the goal, skipping past Petr Sevcik and goalkeeper Ondrej Kolar before rifling home the opener.
The pair combined again in the final minute with another rapid counter, the Germany midfielder this time slipping Hakimi in down the left and the 20-year-old — on loan from Real — slotted a neat left-foot finish that made sure of the win.
"Hakimi sat on his motorbike and raced away from everybody," cooed Slavia defender Ondrej Kudela.
Ajax took control of Group H with a big 3-0 success at Valencia that put them top with a perfect six points and showed again that they could yet repeat last season's heroics.
Hakim Ziyech opened the scoring for the Dutch champions with an incredible dipping strike on just eight minutes, and Quincy Promes doubled their lead 11 minutes before half-time with his second in as many Champions League games before the break, punishing Daniel Parejo for smashing a penalty high over the bar.
Donny van de Beek sealed the points for last season's semi-finalists when he finished a beautiful passing move in the 67th minute, collecting Dusan Tadic's brilliant through ball and slotting home the third.
Erik ten Hag's side are three points clear of the La Liga side and Chelsea, who won a tough encounter at Lille thanks to Willian's 78th-minute volleyed winner that leaves the Ligue 1 outfit bottom without a point after two defeats.
Tammy Abraham had given Frank Lampard's side a deserved 22nd-minute lead with his first ever Champions League goal before more slack set-piece defending allowed Victor Osimhen to level not long after.
Zenit St Petersburg top Group G after seeing off Benfica 3-1 in Russia, level on four points with Lyon after their 2-0 win at RB Leipzig.