Luis Diaz started for Liverpool hours after his kidnapped father was released but lost 3-2 to Toulouse on Thursday as West Ham ensured they have European football in 2024 and Leverkusen progress.
The stunning upset in France came courtesy of goals from Aron Donnum, Thijs Dallinga and Frank Magri as Liverpool sank to their first defeat in the Europa League this campaign.
However, the game was not without controversy with VAR intervening to rule out an equaliser deep in injury-time from Jarell Quansah.
"It was well deserved to lose because they won pretty much all the decisive challenges, battles," said Liverpool boss Jurgen Klopp as his side missed a chance to book their place in the knock-out stages.
Toulouse now sit on seven points, two behind the Premier League club and three ahead of third-placed St Gilloise in Group E.
With 36 minutes gone, Donnum pounced on Kostas Tsimikas and robbed him of possession before slotting home via a deflection to give Toulouse the lead at the break.
The buoyant hosts got a second when Dallinga took a touch in the box and drilled past Caoimhin Kelleher on 58 minutes.
An own goal from Cristian Casseres on 73 minutes gave Liverpool a lifeline but the hosts, pushed on by a vibrant home crowd, punched back immediately.
Three minutes later, Kelleher could not hold a firmly hit cross and Magri was on hand to restore Toulouse's two-goal buffer.
Diogo Jota, on for Diaz, struck a second for Liverpool in the 89th minute.
The hosts were able to see the game out to claim a precious three points, but not before an equaliser seven minutes into injury-time by Quansah was ruled out after video review for an adjudged handball much earlier in the build-up.
- West Ham's journey continues -
Last year's Europa Conference League winners West Ham ground down Olympiakos with a Lucas Paqueta goal on 73 minutes being the difference between the two sides.
"I'm really pleased that we got the result. We didn't play as well as we'd have liked but Olympiakos are a decent team," said West Ham manager David Moyes to TNT Sports.
The win moves them top of Group A and with nine points they are assured of either Europa League or Europa Conference League football in the new year.
Villarreal beat Maccabi Haifa 2-0 in Larnaca, Cyprus with the fixture moved due to the Israel-Hamas conflict.
Rennes ran out 3-1 winners against Greece's Panathinaikos, while fellow French side Marseille went top of Group B with a 2-0 win at AEK Athens.
In Group B, Brighton took a big step towards booking their place in the knock-out stage with a comfortable 2-0 win away to Ajax.
Goals either side of the break from Ansu Fati and Simon Adingra gave the Seagulls a famous win on the home turf of European royalty.
The Premier League team now have seven points and sit three clear of AEK Athens in third place.
Rangers gained a crucial three points over Sparta Prague with a 2-1 home victory thanks to Danilo and Todd Cantwell.
Bundesliga leaders Bayer Leverkusen wrapped up qualification for the knock-out stage with a 1-0 win in Baku against Group H rivals Qarabag.
A Victor Boniface penalty in the fourth minute of second-half injury-time was enough to give Xabi Alonso's side a fourth win in four.
Alongside Atalanta who won 1-0 against Sturm Graz, Leverkusen have booked their place in the next stage of the competition, ensuring they will at least reach the knock-out round play-offs.
- Slavia sack Rome -
The second biggest surprise of the night came in Prague when Slavia Prague downed Jose Mourinho's Roma 2-0 to leapfrog the Italians into first place in Group G.
"I can't explain this. I don't know myself," said Mourinho. "I'll be honest -- Slavia extremely deserved to win and Roma extremely deserved to lose."
Strikes from Vaclav Jurecka in the 50th minute and Lukas Masopust 16 minutes from time inflicted Roma's first defeat of the campaign and moved Slavia to top spot with nine points, level with the visitors.
In the group's other game, Swiss side Servette beat Sheriff Tiraspol 2-1 thanks to a 92nd minute penalty by Chris Bedia.