Benfica beat five-time European champion Liverpool 2-1 to move a step closer to the Europa League semifinals, while Fulham stunned German champion Wolfsburg by the same scoreline on Thursday.
Wolfsburg's German rival Hamburg enjoyed a better night, beating Standard Liege 2-1 at home as it chases a spot in the May final at its own stadium.
In the all-Spanish contest, Atletico Madrid scored two vital away goals in a 2-2 draw at Valencia.
The second legs will be played next Thursday.
Daniel Agger did score a crucial away goal in the quarterfinal first-leg for Liverpool, which had Ryan Babel sent off before Benfica's Oscar Cardozo scored two second-half penalties.
Liverpool's crosstown rival Everton was condemned to a 5-0 loss at two-time European champion Benfica earlier in the season, but Agger netted after nine minutes for Rafa Benitez's side, which has been struggling in the English Premier League and was eliminated early from the Champions League.
Captain Steven Gerrard squared a clever free kick along the ground into the penalty area, where the unmarked Agger sent a clever backheel into the corner of the net.
But Liverpool's hopes were dealt a severe blow when Babel was dismissed in the melee following Luisao's foul on visiting striker Fernando Torres. Babel placed his hand over Luisao's face and thrusted it again toward the defender, leading Swedish referee Jonas Eriksson to brandish the red card.
Luisao was only booked, angering Benitez.
"I was surprised because a foul from behind is more serious," Benitez said. "It is a big difference when you kick a player from behind to a player touching another in the mouth. It is difficult to understand and impossible to change."
Liverpool's fortunes took a further turn for the worse after the break when Emiliano Insua, whose first-half booking for pulling back Di Maria rules him out of the second leg, brought down Aimar in the penalty area.
Cardozo buried the spotkick into the bottom right of the net - as he did 20 minutes later following Jamie Carragher's handball.
"We could have scored more but it was not an easy game," Benfica coach Jorge Jesus said. "It was easy after the sending-off but we played well, we were more offensive after that and the two best teams in this competition were playing against each other."
While Liverpool is underachieving this season, Fulham is confounding all expectations in the Europa League.
Bobby Zamora's curling strike in the 59th minute put Fulham ahead in its first ever European quarterfinal. And Zamora set up Damien Duff for the second four minutes later.
But Alexander Madlung headed in a cross in the 89th to raise Wolfsburg's hopes of a second-leg fightback.
"We did enough to show we can cause them plenty of problems when we attack," Fulham manager Roy Hodgson said. "I thought we played very well and were close, at the end of the game, to a famous 2-0 victory.
"The late goal, I suppose to some extent, has put a slight dampener on the performance."
In the Hamburg stadium where the May final will be contested, Standard stunned the hosts in the 31st when Dieudonne Mbokani headed home a corner.
But Mladen Petric equalized 11 minutes later for Hamburg from the penalty spot after Marcos Camozzato fouled Jonathan Pitroipa. And the German side went ahead just before the break when Ruud van Nistelrooy struck from 10 yards (meters).
In the 41st all-Spanish matchup in Europe, Valencia twice came from behind to deny Atletico Madrid an away win.
Sergio Aguero raced clear on Valencia goalkeeper Cesar Sanchez and fed Diego Forlan, who tapped the opener into an open goal just before the hour mark.
Manuel Fernandes equalized for Valencia with a powerful shot from outside the area eight minutes later, then Atletico went back ahead in the 72nd when Antonio Lopez headed home Alvaro Dominguez's flicked-on corner kick.
David Villa had a handful of opportunities saved before scoring his 24th of the season to even it for good in the 82nd. The Spain striker latched on to substitute Vicente Rodriguez's centering pass to beat goalkeeper David De Gea with a quick shot.
