The Bavarians showed a strong battling spirit to come from behind twice at the Trolli Arena, but had no answer to Hannover's third goal as they remained 12 points from safety with four games of the season to go.
Mohammed Abdellaoue and Andre Hoffmann twice gave Hannover the lead with Nikola Djurdjic responding each time.
Sergio Da Silva Pinto then scored the winner in the 87th minute to take Hannover up to ninth in the table and still in with a small chance of qualifying for Europe next season with just four points separating them from sixth-placed Eintracht Frankfurt.
Furth showed last weekend how, despite their seemingly inevitable relegation to the second division, they are keen to cause a few upsets on their way.
Having beaten local rivals Nurnberg on their own patch, the aim on Friday night was to notch their first home win of the season, and they soon made their intentions clear.
Felix Klaus' shot from a tight angle went wide before Edgar Prib wasted a glorious opportunity to give the hosts the lead, lifting his shot over the bar with only Ron Robert Zieler to beat.
Stephan Furstner then had two shots blocked as the home side maintained a constant pressure on their guests, who were struggling to get out of their own half at times.
With their first dangerous attack of the game, though, they took the lead through Abdellaoue.
He ghosted in for an easy tap-in after Didier Ya Konan's low shot evaded everybody in the middle, including Lasse Sobiech's attempt to clear.
Furth did not let the goal knock them out of their rhythm and they continued to probe, getting their reward within five minutes.
Djurdjic exchanged passes with Klaus before beating Karim Haggui and sending in a low shot past Zieler.
Even in the second half, Furth seemed closer to a second goal before Hannover regained the lead against the run of play.
Konstantin Rausch's corner was met by the head of Hoffmann, who nodded the ball in low to the right of Wolfgang Hesl.
Furth responded well again and, for the second time in the match, got a deserved equaliser through Djurdjic with a close-range header from Johannes Geis' free-kick.
Sensing the opportunity of their first ever home win in the Bundesliga, Furth threw caution to the wind, but were caught out three minutes from time when Da Silva Pinto grabbed a winner for the Reds, leaving Hesl no chance from Mame Diouf's pass.