Japan's Muto scores first Bundesliga hat-trick
The 23-year-old has now scored six goals in 11 Bundesliga games since signing for Mainz at the start of the season.
Muto gave Mainz a 2-0 lead before Augsburg roared back with three unanswered goals, but the Japanese saved his side's blushes with virtually the last kick of the game.
Muto, who chose Mainz over Premier League side Chelsea, opened the scoring when Pablo De Blasis picked him out on the edge of the six-yard area to fire home on 18 minutes.
The ex-FC Tokyo star, who has made 17 appearances for Japan, made it 2-0 with another close-range finish on the half-hour mark, but Paul Verhaegh's penalty for Augsburg made it 2-1 at half-time.
The hosts drew level when South Korea international Koo Ja-Cheol headed home on 50 minutes, then Raul Bobadilla put the hosts ahead with ten minutes left.
But Muto bundled home his third with the last kick of the match to earn the visitors a share of the spoils and the result kept Augsburg bottom of the table with Mainz 13th.