All four teams in action on Sunday had lost their opening two games and further defeats left Braunschweig, on their return to the top flight after an absence of 28 years, and Stuttgart as the only two of the Bundesliga's 18 teams with no points.
Stuttgart completed a miserable day by having Ibrahima Traore sent off only six minutes after coming on as a second-half substitute.
Frankfurt dominated the first half, Takashi Inui hitting the post after 10 minutes, although they had a let-off when Simeon Jackson intercepted a back pass but saw his effort saved by Frankfurt goalkeeper Kevin Trapp.
The Eagles went ahead early in the 52nd minute when a shot by Alexander Meier, making his 250th league appearance for the club, took a wicked deflection off Ermin Bicakcic's shoulder and flew in at the near post. The goal was awarded to Meier, making it the 50th he has scored in the Bundesliga.
Ten minutes later Braunschweig lost possession as they tried to play the ball out of defence.
Omar Elabdellaoui's pass across the face of the penalty area was intercepted by Stefan Aigner, who ran on to fire a low, right-foot shot past the helpless Daniel Davari.
Augsburg, who avoided the drop by three points last season, were quickly off the mark when Halil Altintop took advantage of sloppy defending to score from close range in the sixth minute.
Stuttgart's defence was again caught napping in the 36th minute, leaving Jan Ingwer Callsen-Bracke unmarked to head in Raphael Holzhauser's free kick from the right.
Stuttgart's troubles were not over, however, and, after coming on in the 64th minute, Traore was given a straight red card six minutes later for a foul on Ronny Philp in what seemed a harsh decision.