The evening started well for the Bavarians when Nikola Djurdjic headed them in front in the second minute, but goals from Takashi Inui, Stefan Aigner and Alexander Meier turned the game around.
Sercan Sararer brought Furth back to within a goal, but despite their efforts, they were unable to find an equaliser as they remain rooted to the foot of the Bundesliga, 14 points away from safety with seven games to go.
Frankfurt's win takes them up into fifth place, level on points with fourth-placed Schalke and still in the hunt for a place in the Champions League next season.
Furth, without a win at home all season, made an ideal start with Djurdjic heading in Bernd Nehrig's deflected cross at the far post.
The lead lasted only 10 minutes, though, before Inui picked his spot low in the bottom left-hand corner from Meier's assist to bring Frankfurt level.
The goal gave Frankfurt more confidence and they had chances to take the lead before the interval as Furth were forced to defend.
They finally went in front in the 58th minute when Aigner's shot deflected off Rahman Baba to nestle in the top right-hand corner of the goal.
Ten minutes later, Meier ended his 800-minute goalscoring drought to net his 13th goal of the season from outside the penalty area and give the visitors a two-goal cushion.
It was Furth's turn to respond, and they did so with Sararer reducing the arrears in the 72nd minute with a shot which went in off the inside of the post.
The other side of the post prevented them from levelling a minute later when Djurdjic went agonisingly close to his second of the game.
The Bavarians continued to push for a point, but failed to find one as Frankfurt packed men behind the ball to preserve a victory which takes them back into the top six.