Goals from Lewis Holtby and Klaas-Jan Huntelaar late in the first half turned the game around for the visitors after Kaiserslautern, who have not won in 16 consecutive games, had taken a third-minute lead through Rodnei.
Spaniard Raul then ended any lingering hopes the hosts may have had when he rifled in from 19 metres with an unstoppable left-footer for his 12th goal of the league season and Jefferson Farfan tapped in another nine minutes from time after a quick break involving Holtby.
"It would great if we could finish fourth this season," said Schalke coach Huub Stevens. "We said we wanted to get a quick third goal after the break and after Raul's superb goal everything was easy."
The top three Bundesliga finishers qualify for the Champions League group stage while the fourth team plays in the competition's preliminary round.
"If we want to stay up we have to play differently than we did today," Kurz told reporters. "Now we need to get back up and we will get back up."
January signing Mame Diouf scored twice after goals from Lars Stindl and Jan Schlaudraff. Cologne's Kevin Pezzoni had briefly levelled just before the break.
