Edu put Furth ahead in the eighth minute before Nils Petersen equalised in the 44th minute for Bremen.
Despite the point, Furth dropped to bottom with six points while Werder moved to 11 points.
Furth took an early lead as Edgar Prib played a through ball to the left side to Zoltan Stieber who crossed to the far right post where Edu slotted home.
Prodl appeared to have equalised two minutes later but committed a foul in the build-up to his headed goal.
Bremen continued to dominate, but Marko Arnautovic's shot was stopped by Furth goalkeeper Max Grun in the 24th minute and he also held Petersen's shot from eight yards four minutes later.
Zlatko Junuzovic's 19-yard blast was deflected just inches wide right by Petersen in the 31st minute.
The visitors equalised a minute before the break as Austrian internationals Arnautovic and Junuzovic played a one-two before Arnautovic crossed back into the box where Petersen headed home from five yards out.
Werder had another chance five minutes after the break but Petersen shot over the bar.
Furth failed to convert a three-on-two counter attack as Stieber decided against shooting and passed to Prib, who also chose to pass and Gerald Asamoah was ruled offside.
Bremen fashioned two chances in the 59th minute but Grun saved Petersen's attempt and Arnautovic's rebound shot as well before Heinich Schmidtgal cleared the ball
Furth had another opportunity in the 68th minute but Thomas Kleine headed wide from nine yards out.
And the hosts were close one final time in the 84th minute but Schmidtgal's blast found only the side netting.