Violence broke out before the game, which Hannover won on penalties following a 1-1 draw after extra-time, as some 300 Dynamo fans broke into the stadium and clashed with police.
Dresden fans then stormed the pitch after their team's defeat, again prompting police intervention.
In total, there were nine injuries while 18 fans were taken into custody.
Only the presence of 1,000 police officers kept large groups of rival fans apart.
Dresden fans were in hot water almost a year ago after rioting at Borussia Dortmund, which prompted the Federation to slap the eastern club with a €100,000 fine.
Sixteen people were arrested and several hurt on Wednesday, including police officers and three Dynamo supporters, at Hannover's AWD Arena stadium.
"We were able to deal with the situation but unfortunately a number of Dynamo fans did not follow police instructions and did nothing to improve their bad image," said Hannover police chief Bernd Kirschning.
With fans chanting hate slogans, burning flares and throwing bottles, one unnamed police officer told Bild newspaper, "I have never seen so much aggression at a game. For the first time on duty, I feared for my life".
"The images are shocking," Andreas Rettig, the chief executive of the German Football League (DFL), told broadcaster ZDF after seeing pictures of the violence.
"If that is the perception of a police officer, then that is dramatic. Visiting a stadium is supposed to be safe and pleasant for players, referees and fans. This is not a good development."
Violence at football grounds is a recurring cause for concern in Germany. The last major incident was on October 20, when 180 people were arrested after Ruhr rivals Schalke and Dortmund fans clashed on the pitch.
The DFL are planning a security concept - Secure Stadium Experience - but several points are proving controversial with the clubs, including proposals for full-body searches and lengthy bans for some fans.
"The paper is not set in stone and will be fully discussed with all stakeholders," said Rettig with clubs holding a summit in Berlin on Thursday.
He added: "99 percent of fans are not involved, because they are peaceful.
"The Herculean task is to separate the fans so that not everyone is sanctioned."