After laying siege to their hosts' goal without scoring in the first half, Frankfurt finally went ahead through Alexander Meier in the 52nd minute.
Ten minutes later, Stefan Aigner sealed the first win of the season for Armin Veh's side as Braunschweig suffered a third defeat in as many top-flight games following promotion.
Frankfurt made their intentions clear immediately, pressing Braunschweig back into their own half and making it seem more like a home game.
Takashi Inui was posing the greatest threat and his frequent incursions from the left contributed to Frankfurt creating more chances in the first 30 minutes than in the entirety of either of their previous two games.
Inui's had the first shot of the game, which was deflected for a corner, from which Sebastian Rode went close with a header.
Inui then hit the outside of a post after Rode's shot was parried out by Daniel Davari, who was doing all he could to keep the league newcomers on terms.
He succeeded in doing so until half-time, but he could do nothing to prevent Meier from giving the visitors the lead seven minutes into the second half.
Frankfurt's top goalscorer both this season and last got lucky with a shot taking a wicked deflection off Ermin Bicackic, completely wrong-footing Davari as it looped inside his near post.
While the first goal could be put down to misfortune, the second was a clear mistake with Omar Elabdellaoui playing an unnecessarily dangerous ball across the face of his own goal.
Aigner got to it before its intended recipient and he coolly slotted a shot low past Davari to double Frankfurt's lead.
Braunschweig tried to respond with Domi Kumbala sent on by coach Torsten Lieberknecht.
He had three chances to mark his Bundesliga debut with a goal, but missed each time, summing up Braunschweig's day and their ponderous start to the season.