The 26-year-old has already spent three years in Germany's top flight having made 59 appearances for Borussia Moenchengladbach, scoring eight goals between 2009 and January 2012.
Bobadilla was under contract at Basel until 2017, but was suspended by the club in July after being stopped by police having driven his Maserati sports car at 111 kilometres per hour through a village where the limit was 50km/h.
Basel signed the Argentinian from Swiss league rivals Young Boys in January while he was still completing a seven-match ban for insulting a referee after being shown a red card.
Augsburg coach Mark Weinzierl said he hopes Bobadilla will strengthen their attack, which failed to score in last Saturday's 4-0 defeat at home to Champions League finalists Borussia Dortmund on the opening weekend of the Bundesliga season.
"I am grateful to FC Augsburg for giving giving me the chance to play in the Bundesliga again," Bobadilla said.
He is the third Basel player to be traded to German team in the last few months.
As a replacement for Bobadilla, Basel are reported to be interested in signing Hoffenheim's Eren Derdiyok and Fulham's former Hamburg striker Mladen Petric.