Frankfurt looked firmly in control after Jonathan Burkardt headed them into an early lead, but Augsburg turned the game around immediately after half-time with two goals in less than two minutes.
The result gives Augsburg their best start to a Bundesliga campaign since they were promoted in 2011 and puts them top of the table for the first time in the club's history.
Frankfurt dominated much of the first half and could have extended their advantage, with Can Uzun, Raphael Onyedika and Younes Ebnoutalib all testing goalkeeper Finn Dahmen.
Dahmen was particularly important when he denied Ebnoutalib in a one-on-one situation, keeping Augsburg within touching distance at the break.
Augsburg had offered little going forward, but Fabian Rieder came close with a stoppage-time free-kick. That chance proved to be a warning of what was to come.
Just 26 seconds into the second half, Marius Wolf sent a high cross into the box and Arijon Ibrahimovic rose to head Augsburg level, with the ball taking a deflection off Elias Baum.
Augsburg then struck again almost immediately. Ibrahimovic broke forward from the restart and forced Noah Atubolu into a mistake, with Robin Fellhauer following up to bundle the ball home.
The comeback was complete after the hour mark when Alexis Claude-Maurice found space on the edge of the area before curling a superb finish into the side netting.
Frankfurt manager Adi Hutter made a series of changes in an attempt to regain control, but Augsburg continued to look the more dangerous side.
Rieder then had the final say in the closing stages, making up for his earlier miss by curling a superb free-kick over the wall and into the top-right corner.
For Ibrahimovic, it was a dream start to life at Augsburg after joining on loan from Bayern Munich.
"It was a great feeling to stand in front of the fans after the game and celebrate the win," the 20-year-old said.
"It doesn't get any better than that. It was a brilliant performance from the whole team, even if the first half wasn't our best.
"We believed in ourselves, and I'm proud and happy. You could see how united we are as a team."
Augsburg coach Manuel Baum admitted his side had been fortunate to go into half-time only one goal behind.
"That's the beauty of football. Nobody would have expected that scoreline after the first half," Baum said.
"We should thank Finn for the fact that we were only 1-0 down at the break.
"We were too timid and Frankfurt had quite a few chances. But this team believe in themselves and you could see that after the break."
Baum also praised Augsburg's substitutes for helping his side maintain their momentum.
"Of course, there was also an element of luck involved with the quickfire goals. The key was that we kept on believing, and the players who came off the bench made a real impact," he said.
"That's what this team is all about."
The defeat was another setback for Frankfurt, who had also surrendered a lead in their previous match. Their seven-game unbeaten home record against Augsburg has now ended, while they have won only six of their last 30 competitive matches.