The Dutchman led the way for the Gelsenkirchen side as they came back from the brink of elimination to progress to the last eight of the continent's second-tier club competition
Klaas-Jan Huntelaar was the hat-trick hero for the hosts, as his three goals brought his tally to 37 in all competitions this season, exceeding Klaus Fischer's club record.
The Eredivisie side, having won the first leg 1-0 in Netherlands, scored a critical away goal early through Willem Janssen. Huntelaar equalised on the half hour mark, however, and made it 2-1 on 57 minutes as he converted from the penalty spot. Jermaine Jones scored a critical third goal with 19 minutes left to play, and Huntelaar put the tie to rest on 81 minutes as he completed his hat-trick.
Jones was first to threaten on goal as his long-ranged effort forced a nice save from Nikolay Mihaylov. But in the 14th minute, Janssen stunned the home side as he finished a sparkling passing sequence between Luuk de Jong and Nacer Chadli with a brilliant strike from 18 yards.
Chadli almost put the tie to rest shortly thereafter, but struck well wide and over the bar after being spotted unmarked in the box.
Schalke later improved, and equalised on 29 minutes as Julian Draxler's cross fell to the unmarked Huntelaar, who nodded home from close range. The hosts pushed for a second goal before half-time, but were unable to find the net and instead were found over-committed to offence as Twente counter-attacked. Before the interval, De Jong and Raul had chances saved at either end.
Schalke came out firing in the second half as Draxler and Raul came close in rapid succession. De Jong replied at the other end with a well-struck attempt from outside the box, but his effort - a rare opportunity for the Dutch side - sailed wide.
The hosts maintained their pressure, and caught a break shortly before the hour mark as Janssen blocked Raul's volley with his hand, resulting in a penalty. Huntelaar stepped up to the spot and powered home his finish.
Still in need of one goal in order to progress, S04 pressed forward and made it 4-1 with 18 minutes left to play. Raul played a clever back-heel to Jones, who turned and fired in from close range.
The USA international Jones turned provider shortly thereafter, as he led a counter-attack, and played in Huntelaar, who cut to the inside and struck home with his left foot, putting the tie out of reach.
The result sees Schalke progress to the last eight of the Europa League, where they will face a to-be-determined side that will be announced during Friday's draw in Nyon.