Sascha Molders scored twice with Ja-cheol Koo also on target for the Bavarians, who moved three points clear of the foot of the table and to within a point of Hoffenheim, who currently occupy the Bundesliga's relegation play-off berth.
Stefan Reisinger scored two late consolation goals for Fortuna, who must now look over their own shoulders with the relegation zone closing in on them.
It was clear from the moment Daniel Baier unleashed a powerful shot from distance in the second minute that this Augsburg side was not the same one which ended 2012 with just nine points from 17 games.
They never allowed Fortuna to settle inside their own back yard and created chance after chance with Ronny Philp looking inspired on his debut.
New signing Ji Dong-won was also making a nuisance of himself and he almost marked his debut with a goal in the 38th minute, but Fabian Giefer stretched to deny him.
Two minutes later, Molders gave Augsburg the lead when he took advantage of some confusion between Giefer and Juanan to score into an open goal.
Koo, whose form in the second half of last season was key to Augsburg avoiding relegation, made it two on the stroke of half-time with a shot from a tight angle after being set up by Tobias Werner.
Fortuna tried to make a game of it in the second half, but Augsburg would not be budged and they increased their lead in the 71st minute.
Luck played a big part as Giefer's clearance hit Molders and cannoned into the goal.
Fortuna reduced the arrears just two minutes later with Reisinger finishing off a move which was created by Robbie Kruse's run.
Fortuna pushed for a second, but they had to wait until stoppage time before they found it with Reisinger again on target, turning Oliver Fink's header in at the far post.
There was not enough time for them to find an equaliser, though, as Augsburg held on to claim only their second win of the season.