Napoli was one of four teams to reach the knockout stages of the Europa League in Thursday's matches, cruising into the last 32 with a 5-0 victory over FC Midtjylland.
Molde, Borussia Dortmund and Rapid Vienna also progressed with two games to spare but Schalke missed the chance to join them as it drew 1-1 at Sparta Prague, which remained second in Group K.
Napoli also secured top spot in Group D.
"I have worked hard but I have to thank the players who have followed me, because it's not a given that you arrive from a provincial team and you're followed by players from a great team," said Napoli coach Maurizio Sarri, who joined from Empoli in the offseason.
"I have to also thank the president who strongly defended me even when the results weren't coming."
Teenager Jordon Ibe fired Liverpool to its first victory in the competition, a 1-0 success at Rubin Kazan, as coach Juergen Klopp made it three wins in a row after starting his time at the Premier League club with three consecutive draws.
Liverpool remained second in Group B, but moved to within two points of Sion, which drew 1-1 against Bordeaux and missed a chance to seal its progression.
Midtjylland had won its first two games but lost 4-1 at home to Napoli and looked set for another hefty defeat when Omar El Kaddouri fired the hosts into the lead in the 13th minute with his first goal for the club.
Lorenzo Insigne set up that goal and also had a hand in the second 10 minutes later, with an effort which Joachim Andersen parried into the path of Manolo Gabbiadini, who turned in the rebound.
Gabbiadini doubled his tally in the 38th and Christian Maggio scored his first goal since January 2013 after the break — and his first in Europe in seven years. Jose Callejon headed in the fifth.
Napoli has won all four Europa League matches this season, scoring 16 goals and conceding just one. Midtjylland remained second, two points ahead of Club Brugge, which beat Legia Warsaw 1-0.
Group G leader Lazio needs just one point to join its fellow Italian side in the last 32 after Filip Djordjevic scored both goals in a 2-0 win at Rosbenborg. It remains three points ahead of Saint-Etienne, which beat Dnipro Dnipropetrovsk 3-0 to leave the Ukrainian side six points off the pace.
Serie A leader Fiorentina won 2-0 at Lech Poznan — with Josip Ilicic netting twice — to leapfrog into second in Group I, three points behind Basel.
Dortmund also eased into the next round, beating Qabala 4-0 with Pierre-Emerick Aubameyang — who netted a hat trick against the Azerbaijani side last time — scoring the second.
Rapid Vienna won 2-1 at Viktoria Plzen to advance, while Villarreal needs just a point to also progress from Group E after winning by the same scoreline at Dinamo Minsk, which saw its qualification hopes ended by a fourth defeat.
Molde won 2-1 at Celtic to progress from Group A after Ajax was held to a goalless draw at home by second-place Fenerbahce.
Liverpool was looking for a first victory in the competition after three successive draws and James Milner hit the crossbar in the sixth minute.
Klopp's side dominated and took a deserved lead seven minutes into the second half, when the 19-year-old Ibe raced unchallenged toward the area before firing in off the far post.
Harry Kane continued his fine form to set Tottenham on its way to a 2-1 against Anderlecht, with his fifth goal in three games.
Substitute Mousa Dembele scored a stunning winner three minutes from time to help the English side move top of Group J, a point clear of Monaco which rescued a 1-1 draw at FK Qarabag.
Braga's hopes of qualifying on the day were dashed by a 1-0 defeat at Marseille, dropping its first points of the competition.
It was the second setback faced by the Portuguese club in France. The entire squad's boots were stolen before the match — most likely on Wednesday night.