Two minutes later Pirmin Schwegler smashed his shot against the left post for Frankfurt from a direct free kick from distance.
Moments later the hosts went ahead when Sascha Molders put the ball inside his own net.
Augsburg pulled one back on the stroke of half-time through Koo.
Seven minutes into the restart Frankfurt restored their two-goal advantage, Alexander Meier turning in Stefan Aigner's pass from the centre of the box.
Augsburg came back into the game on 64 minutes when Molders redeemed himself for his earlier own goal by making good use of Torsten Oehrl's pass.
However, Frankfurt were in no mood to allow their visitors to stage a comeback and scored their fourth goal of the afternoon with 15 minutes left on the clock.
Matthias Ostrzolek was adjudged to have handled the ball inside the Augsburg area, and Meier successfully converted the spot kick.