Leistner, 30, was giving a post-match interview on Monday after his team were beaten by former club Dynamo when he was taunted by members of the crowd at Glücksgas Stadium.
The footballer was filmed scaling a concrete wall and then charging up the steps to confront one man in a maroon T-shirt in the stands.
Tensions were eventually defused and Leistner later took to Instagram to apologize, but revealed he had been subjected to “below-the-belt” abuse directed “against my family, my wife and my daughter.”
The German Football Association (DFB) has now slapped the defender with a five-match ban – two games of which are suspended for a year – as well as an €8,000 ($9,500) fine.
The federation also censured Leistner for supposedly exposing himself and third parties to the risk of infection with the coronavirus by attacking the fan.
“We consider this to be serious sporting misconduct, so the ban also applies to league operations and not just the DFB-Pokal,” a statement read.
“On the one hand, we had to take into account that an attack on a spectator was particularly serious misconduct by a player, but on the other hand we also had to appreciate the seriousness of the previous provocation.
“The player also said he has not yet appeared in court, regrets his misconduct and has sincerely apologized.
“The additional fine results from the violation of the hygiene concept, which we do not want to trivialize.”
Leistner played for hometown club Dynamo between 2010 and 2014, before moving on to clubs including Queens Park Rangers in England and then arriving at Hamburg earlier this year.
Dynamo described the fan’s behavior at Monday’s cup match as “shameful”.
The game in Dresden saw 2. Bundesliga team Hamburg take on Dynamo, who reside one tier below them, and was attended by 10,000 fans after authorities relaxed restrictions to allow stadiums to be partially filled.
Leistner’s attack was reminiscent of an incident in England last season when Tottenham Hotspur player Eric Dier entered the crowd to confront a supporter for allegedly making derogatory comments about his brother.