With leaders Bayern Munich at second-placed Borussia Dortmund in Saturday evening's top-of-the-table clash, Leverkusen took the chance to trim the gap above them in the table to three points.
Schick became the first Leverkusen player in the club's history to score four goals in a single Bundesliga match.
Hosts Leverkusen gave a masterclass in finishing by scoring seven goals from their first ten chances.
They converted their first chance when French striker Amine Adli hit the bottom corner with just 12 minutes gone. Teenage midfielder Florian Wirtz then floated in a free-kick which was headed home by Edmond Tapsoba with the centre-back completely unmarked in the area.
Fuerth extended their German league record run to 12 straight defeats but pulled a goal back Jeremy Dudziak profited from a mistake in the Leverkusen defence.
Schick was then involved in the next five Leverkusen goals as the hosts ran riot.