Two goals from Adam Szalai and a Yunus Malli strike earned Thomas Tuchel's side all three points as Furth - winless at home this season - fell apart in the second half.
The Bavarians held their own during the first 45 minutes but never looked close to landing an historic first home win in the Bundesliga and remained rooted to the foot of the table on just nine points from 19 matches.
Mainz, on the other hand, can dream of European football next term after climbing to fifth spot.
Furth repeated their all-too-familiar tactics of sitting deep and trying to keep things tight at the back.
Mainz tried to break them down, but had little success in the first period.
Marcel Risse was brought on by Tuchel during the interval and the tactical change allowed them to finally make the breakthrough in the 53rd minute.
Andreas Ivanschitz's shot was parried by Wolfgang Hesl but the ball landed kindly for Szalai, who scored with the aid of a deflection off Rahman Baba.
The floodgates then opened with Malli doubling Mainz's lead in the 65th minute.
Szalai's effort was blocked on the goal-line by a wall of Furth players, but the rebound was again unfortunate from the home side's point of view as Malli picked up possession and threaded his shot in.
Hesl prevented Risse from making it three in the 72nd minute, but he was powerless to deny Szalai 11 minutes later.
The Hungarian was put through by Risse and he waited for Hesl to commit before lifting the ball over him and into the back of the net.