Reus, who was unlucky to miss Germany's World Cup-winning campaign after being injured in their final warm-up game, showed he was back at his best on the day he was recalled to the Germany squad for international matches next month.
For last season's runners-up Dortmund, who are reportedly close to re-signing Japan international Shinji Kagawa from Manchester United, it was their first league win after their 2-0 opening defeat by Bayer Leverkusen.
"I am happy to be playing again and helping the team," Reus told reporters after scoring his first goal since suffering an ankle injury in June.
"We still have a lot of work to do, that was visible late in the second half. When you drop your level a few percent then you will be punished. So, the work continues," Reus said.
Dortmund substitute Adrian Ramos notched his first league goal for his new club, tapping in from close range 12 minutes from time before the hosts struck twice late in the game.
Despite several absentees including captain Mats Hummels and Marcel Schmelzer, Dortmund coach Juergen Klopp saw keeper Roman Weidenfeller, one of their World Cup winners, play his first game this season and deny the hosts late in the first half.
Colombian Ramos, who started on the bench along with fellow striker and 18 million euros (£14.3 million) signing Ciro Immobile, scored from close range five minutes after coming on following a Marwin Hitz save as Dortmund seemed to be cruising towards an easy win.
But Augsburg, whose club president has warned in a letter to fans of a difficult season ahead, pulled a goal back through Raul Bobadilla in the 82nd before Tim Matavz's 90th minute glancing header set up a nervous finale.