Furth came into the match needing a win to keep survival hopes flickering for another week, and a Jetro Willems opener gave it just the start it needed.
But the lead lasted just three minutes as Schick reacted quickest to flick home a corner. Leverkusen went ahead on 18 minutes, again taking advantage of a defensive mistake when Azmoun closed down goalkeeper Andreas Linde to earn a simple tap-in for his maiden Bundesliga goal.
Paulinho added a third for the visitors when Azmoun flicked over the defence, allowing the Brazilian to shoot past Linde at the second attempt. Substitute Exequiel Palacios rounded things off with a tidy finish from 20 yards.
Leverkusen's win, combined with RB Leipzig's defeat at home to Union Berlin, moved it them up to third, but only a point above Leipzig in fourth and three ahead of Freiburg in fifth.
For Furth, the news is not so good and it can no longer climb out of the bottom two, 11 points adrift of 16th place with three games to go.