After a barren first half, the game burst into life in the second period.
Adam Szalai broke the deadlock on 53 minutes when he fired in from the left side of the Furth penalty box with his left foot.
It became 2-0 to Mainz in the 65th minute when Yunus Malli shot home from close range.
Furth surged forward and created a couple of decent chances, but it was the visitors who scored the game's third goal.
Szalai latched onto Marcel Risse's pass to shoot home from outside the box with six minutes of normal time left on the clock.