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.
He set up Ecuador’s Piero Hincapie just before the break, then fired in four goals in just 37 minutes, with two off his left foot, one off his right and a header.
Schick’s four goals means he has now scored 12 times in 11 games, two more goals than Dortmund star striker Erling Haaland.
Only Bayern goal-ace Robert Lewandowski, who has 14 goals in his 13 games, has scored more this season in Germany’s top flight.
Wolfsburg crashed to a 3-0 defeat at Mainz ahead of Wednesday’s all-or-nothing Champions League clash at home to Lille.
Wolfsburg never recovered from conceding a goal by striker midfielder Jonathan Burkardt after just 68 seconds.
They were 2-0 down after only four minutes when midfielder Anton Stach scored from outside the box.
To compound Wolfsburg’s misery, French defender Maxence Lacroix scored a late own goal.