The groups clashed with bottles, knives and screwdrivers in the parking space of a toy store some distance from the stadium, and the fighting spilled out onto the neighbouring highway, paralysing traffic.
A 25-year-old man died before reaching hospital, a hospital official told AFP. He bore serious head injuries as well as stab wounds.
Police detained at least 13 people, including the driver of a car that drove into a throng of fans when they attacked his vehicle.
Vyron Polydoras, Greece's public order minister, said: "This is a very sad and violent scene. I express my regret."
Riot police have been sent to guard the hospital.
Two cars on the highway were attacked, one of them burnt to a crisp by a firebomb.
One driver told NET state television: "They nearly cracked my head open, If the police had taken longer to intervene they would have burned us alive."
"They nearly cracked my head open - if the police had taken longer to intervene they would have burned us alive"
A witness to the brawl
Another witness told Mega TV: "They were jumping on our car for five minutes, they were asking for our cell phones and stabbed our driver."
Three stores in the vicinity were also damaged, the Athens News Agency (ANA) reported.
Fan violence routinely mars Greek sport, usually centred around football and basketball but occasionally affecting other sports such as water polo and volleyball.
Successive Greek governments have spent years trying to combat the problem, stiffening laws, introducing personalised tickets and promoting an initiative to install surveillance cameras inside stadiums.
But success has been limited.
Last weekend, Greek fans managed to sneak flares into a high-stake football match against neighbouring rivals Turkey, a qualifying game for the 2008 European Championships that the authorities had pledged to monitor closely.
Several of the flares were fired onto the pitch, and the Turkish players were targeted with plastic bottles during the entire game.
A brawl also broke out in one of the stands over slogans chanted by one group of Greek fans.