Trailing 4-2 from the quarter-final first leg, Schalke twice took the lead on the night through Klaas Jan Huntelaar and Raul.
But the home side replied with strikes from Ibai Gomez and Markel Susaeta to secure a 2-2 draw which earned them an aggregate victory and a last-four meeting with Sporting Lisbon.
Cleiton Xavier's second-half penalty miss cost hosts Metalist Kharkiv dear as Sporting Lisbon clung on for a 1-1 draw in Ukraine which sent them through to the semi-finals 3-2 on aggregate.
Leading 2-1 from the first leg in Portugal, Sporting extended their lead through Ricky van Wolfswinkel before Jonathan Cristaldo equalised early in the second half.
Metalist blew their big chance of levelling on aggregate in the 63rd minute when Alexandre Xandao fouled Marko Devic in the box, and Xavier rattled his resulting spot-kick at the legs of Sporting goalkeeper Rui Patricio.
Two goals in as many first-half minutes from Adil Rami sent Valencia on their way to the semi-finals as the La Liga side overturned a first-leg deficit to see off AZ Alkmaar.
Despite suffering a 2-1 reversal in the opening fixture in Holland last week, the Spaniards were on top form with Jordi Alba and Pablo Hernandez adding to Rami's early brace to wrap up an emphatic 4-0 win on the night, and a 5-2 aggregate success.
Atletico Madrid progressed to the semi-finals for the second time in three seasons as second-half strikes from Adrian Lopez and Radamel Falcao were enough to see off Hannover 4-2 on aggregate.
With the tie finely balanced from the first leg in Madrid last week, the game did not open up until after the break when Adrian spun away from three desperate Hannover challenges to slot home his sixth goal of the competition.
On-loan Manchester United striker Mame Biram Diouf gave the hosts hope as he pulled a goal back with nine minutes remaining, but Atletico sealed a 2-1 win on the night and their passage into the last four when Falcao rocketed home a volley from the edge of the area five minutes later.
Atletico will now face Valencia in an all-Spanish semi-final.
Thursday's Europa League Fixtures