The Dutch international striker has now scored 37 goals in 37 games after a breathless night in which the German side lost the opening goal before knocking four past Steve McClaren's unit, who simply could not live with the home team in the second period.
Schalke lost the first leg 1-0 in the Netherlands, but progress 4-2 on aggregate.
Huntelaar cancelled out Willem Janssen's 14th minute opener for the Dutch team when he headed home on 29 minutes from Julian Draxler's delivery.
He added his second of the night from the penalty spot after the tireless Jermaine Jones was felled by Dwight Tiendalli on 58 minutes.
Schalke were dominant and wrapped up their progress when Jones turned in the box to net on 71 minutes before Huntelaar finished off a sweeping move on the break to seal the victory on 81 minutes.
The Spanish side ended the match at the Philips Stadion with 10 men after defender Adil Rami was harshly sent off for tussling off the ball with home skipper Ola Toivonen in the closing moments.
It was Rami who scored his side's vital away goal when he nodded home on 47 minutes after directing Sofiane Feghouli's shot into the net with his head.
Toivonen equalised on 64 minutes when he scrambled the loose ball into the net, but it was not enough to prevent the Dutch side from tumbling out.
2010 Europa League winners
Atletico Madrid (Spain)
Athletic Bilbao (Spain)
Sporting Lisbon (Portugal)
AZ Alkmaar (Netherlands)
Metalist Kharkiv (Ukraine)