Giovanni Van Bronckhorst got off to a perfect start as Rangers manager as a 2-0 win over Sparta Prague secured their place in the last 32 of the Europa League, while Leicester edged towards the knockout stages with a 3-1 win over Legia Warsaw.
But Celtic's chances of progress were ended by a late Bayer Leverkusen rally from 2-1 down to win 3-2 in Germany.
Real Betis, Lazio, Monaco, Galatasaray and Olympiakos also qualified for the knockout stages on Thursday.
Van Bronckhorst watched on from the stands as Rangers suffered a shock 3-1 defeat to Hibernian in Sunday's Scottish League Cup semi-final.
But in his first match since formally succeeding Steven Gerrard, Rangers got the two-goal victory they needed to guarantee second place in Group A thanks to early goals in both halves from Alfredo Morelos.
Leicester will travel to Napoli in two weeks' time knowing a point will be enough to progress after a first half blitz of Legia.
Patson Daka, James Maddison and Wilfried Ndidi all scored inside 33 minutes against the Polish champions to jump from bottom of the group before kick-off to top.
All four sides still have the chance to qualify in the final games of the group as the Foxes lead Napoli and Spartak Moscow by a point with Legia one point further back.
- Celtic have gap to bridge -
Celtic will have to settle for a place in the Europa Conference League in the new year after letting the lead slip in Leverkusen.
Robert Andrich headed the hosts in front before Josip Juranovic equalised from the penalty spot.
Jota's sweet strike 11 minutes into the second half had Celtic on course for their first ever win on German soil.
But Andrich smashed home his second of the game between Joe Hart's legs before Moussa Diaby beat the former England international from the edge of the box.
"It was disappointing for the players as they put in a massive effort, but there’s still a gap between us and them, and that’s what we need to bridge," said Celtic boss Ange Postecoglou.
Betis made sure of second place in the group behind Leverkusen thanks to a 2-0 win over Ferencvaros.
Lazio and Galatasaray will face off in two weeks' time for top spot in Group E after securing qualification in style.
The Italians were 3-0 winners over Lokomotiv Moscow, who are set to lose head of sports and development Ralf Rangnick to Manchester United according to reports.
Ciro Immobile scored two second half penalties before Pedro Rodriguez rounded off the scoring.
Galatasaray hold a three-point lead at the top of the group after ending Marseille's chances of progress in a 4-2 thriller in Istanbul.
West Ham's fine return to European football continued with a 2-0 win over Rapid Vienna despite David Moyes naming a much-changed line-up.
Andriy Yarmolenko and Mark Noble were on target for the English side.
Real Sociedad need to beat PSV Eindhoven at home in two weeks' time to go through after losing 2-1 to Monaco.
Goals from Kevin Volland and Youssouf Fofana secured a place in the last 16 for the French side thanks to winning the group.
PSV were 2-0 winners over Sturm Graz to move two points clear of Sociedad in the fight for second.