In the process, Hamburger SV also set the Bundesliga record for going the longest without a goal before finally managing to score in the new season.
Scoreless after the first five matches of the season, Hamburg needed to score inside the first 24 minutes to avoid beating Bochum's unwanted record from 1979. But it could not find the net until the 58th, when Nicolai Mueller canceled out Haris Seferovic's goal from the 44th.
In the earlier match, captain Paul Verhaegh converted a first-half penalty to give Augsburg a 1-0 victory over Hertha Berlin.
Hamburg dominated possession but could not score in the first half, which ended with Frankfurt taking the lead largely against the run of play.
Timothy Chandler broke through on the right and sent a useful cross that was not cleared by the Hamburg defense and Seferovic pounced at the far post to slot home.
Hamburg stepped up the pressure after the break and finally broke its drought when Mueller picked up a through ball from Lewis Holtby and scored as Eintracht mistimed the offside trap.
The ball went through the legs of goalkeeper Felix Wiedwald, who was making his first start in the Bundesliga after the long-term ankle injury sustained by Kevin Trapp. Wiedwald's previous Bundesliga experience consisted of a combined 17 minutes in two appearances, but he had some good saves in the match.
Piazon, a 20-year-old Brazilian who came on only in the 85th, fired a stunning free kick from nearly 30 meters (yards) to seal Eintracht's victory. The ball soared high before taking a sudden dip to bury itself in the upper corner.
Eintracht moved past Augsburg into seventh, while Hamburg stayed last, still looking for its first win.
In Augsburg, the goal in the 27th minute gave the home side its third win of the season and lifted it to eighth place in the standings. Verhaegh had to leave the match in the 71st, still wobbly 10 minutes after a collision of heads with Hertha forward Ronny. After the match, Verhaegh was taken to a hospital to be examined.
Hertha's only credible chance came early in the match when Salomon Kalou's shot was tipped over by Augsburg goalkeeper Marwin Hitz. Berlin stayed 14th.
"I am angry," Hertha's coach Jos Luhukay said. "We lost because of the referee. That was never a penalty."
Kraft also disputed the decision but Bobadilla said there was no doubt.
"l am not the type that just falls down," said the powerfully built striker.
Djurdjic returned to the starting lineup after being on the bench for three matches.
"He ran a lot and created a lot of space, that was the plan," Augsburg's coach Markus Weinzierl.
Djurdjic said he was just pleased to be playing.
"It's been an exciting week," he said.
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