The South Korean attacker, who has now scored four goals since his January loan move from Wolfsburg to Augsburg, netted the equaliser on 23 minutes.
In the first minute Mario Gomez had put Bayern ahead but Augsburg gradually came into the game and were level when Koo struck.
Axel Bellinghausen broke down the left after Jerome Boateng's error and his central ball found Koo who fired home from the edge of the box with a shot which Manuel Neuer couldn't prevent.
Gomez, though, would once again have the final say, tapping home Arjen Robben square ball for the winner on the hour mark.
In Scotland's Premier League, Ki Sung-Yueng and Cha Du-Ri celebrated clinching the title as Celtic crushed Kilmarnock 6-0 on Saturday. The win gave Celtic an unassailable 18-point lead with five games to play.
Ki, whose set-pieces causes Killie issues all afternoon, played his part in the opening goal too, which came thanks to Charlie Mulgrew on eight minutes.
The South Korean midfielder fired in a powerful drive which was deflected for a corner. From the resulting set-piece, Ki delivered for Mulgrew to head home.
Ki went on to play with full game, as goals from Glenn Loovens, Mulgrew again, Anthony Stokes and a Gary Hooper brace, sealed the win and title. Cha was an unused substitute.
|STAR OF THE WEEKEND
Was his side's main attacking threat again, with a goal at the Allianz Arena, but couldn't prevent a loss
Lee Chung-Yong (Bolton): Out injured (leg) for the 3-0 home loss to Fulham.
Park Chu-Young (Arsenal): Not in the squad in the 1-0 victory over Manchester City.
Park Ji-Sung (Manchester United): Was an unused substitute in the 2-0 win over QPR.
Son Heung-Min (Hamburg): Was an 88th-minute substitute in the 1-1 draw with Bayer Leverkusen.
Yu Tae-Sun (FSV Frankfurt): Was an 80th-minute substitute as FSV blew a three-goal lead to draw 3-3 with St Pauli.
Jung Jo-Gook (Nancy): Was an unused substitute in the 2-0 triumph away to Dijon.
Suk Hyun-Jun (FC Groningen): No Eredivisie action as the Dutch Cup final was on.
Park Joo-Ho (FC Basel): Put in a full shift in the 1-1 draw away to Luzern.
Kim In-Sung (CSKA Moscow): Was an unused substitute in the goalless draw with Anzhi Makhachkala.
Cho Yong-Hyung (Al Rayyan): Put in a full shift in defence in the stunning 8-2 triumph over Qatar SC.
Lee Jung-Soo (Al Sadd): Absent for the 3-1 win over Al Wakrah.
Nam Tae-Hee (Lekhwiya): Absent for the 1-1 draw with Al Kharitiyath.
Yoo Byung-Soo (Al Hilal): Was a 77th-minute substitute in the 2-0 win at Najran.
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