Europe's second-tier competition offers a last chance of silverware for both under-pressure managers Ange Postecoglou (Spurs) and Ruben Amorim (Manchester United), ahead of next Thursday's second legs.
Both teams are at home with Postecoglou's men trailing AZ Alkmaar 1-0, while United and Real Sociedad are locked at 1-1.
A comical first-half own-goal from Lucas Bergvall consigned a lacklustre Tottenham to the loss away to Alkmaar.
Bergvall's sliced clearance from a Troy Parrott shot looped in after 18 minutes and it proved enough to give AZ a first victory over an English side in 12 attempts.
Spurs had eight days to prepare for this fixture but produced a flat display in the Netherlands and did not register a shot on target until the 88th minute.
It was a thoroughly disappointing performance from Ange Postecoglou's team, as the Australian resisted the temptation to start fit-again trio Micky van de Ven, Cristian Romero and Dominic Solanke.
But Postecoglou remains defiant.
"We'll analyse it and have a look at the areas where we struggled with, but it's fair to say it was nowhere near the level it needed to be," Postecoglou said.
"If we address those things I'm sure we'll be in a position to at least cause them more problems that we did tonight. It's only 1-0 so I guess that's a positive in that we didn't let the game get away from us."
Absentee-hit Manchester United were unable to turn a positive performance into victory as Mikel Oyarzabal's penalty saw Real Sociedad head to Old Trafford with the last-16 tie poised at 1-1.
Joshua Zirkzee put United into a deserved lead, only for Oyarzabal to level from the spot with Sociedad's first shot on target after the VAR spotted a Bruno Fernandes handball.
The rejuvenated hosts ended on top as United's lack of options and recent exertions took their toll, but they head back to Old Trafford for next Thursday's second leg in a decent position.
"I felt our team in the last 30 minutes were really, really tired, and you can feel that in the game. So we take this to Old Trafford," said Amorim.
"It is going to be a different game, the pressure is going to be on us in that stadium, and we have to be ready. I think the key will be surviving physically to Sunday and to be ready and fresh for Thursday."
Rangers took a big step toward the quarter-finals by stunning José Mourinho's Fenerbahce 3-1 in Istanbul, while Malick Fofana scored a brace as Lyon defeated FCSB 3-1.
In the late matches, Ajax went down 2-1 to Eintracht Frankfurt at home with the Germans coming from behind to post the win with goals from Hugo Larsson (27th) and Ellyes Skhiri (70th).
Roma, with goals from Angelino (56th) and Eldor Shomurodov (93rd), beat 10-man Athletic Club 2-1, with Yeray Alvaraez sent off in the 85th minute.
Viktoria Plzen lost 2-1 at home to 10-man Lazio, while Bodo Glimt posted a 3-0 win over Olympiacos.
In some of the third-tier Conference League action, Chelsea beat Copenhagen 2-1 with goals from Reece James and Enzo Fernandez.
Panathinakos beat Fiorentina 3-2 at home, while Real Betis and Vitoria SC drew 2-2.
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