Swiss winger Ruben Vargas gave Augsburg the lead with a superb bicycle kick just before half-time.
Mainz sub Karim Onisiwo equalised on the hour mark, but Augsburg took advantage of tired legs as Germany winger Hahn netted twice in the final 10 minutes.
Werder Bremen coach Florian Kohfeldt picked up a point on his third anniversary in the job with a 1-1 draw at Eintracht Frankfurt.
US striker Josh Sargent swept his shot home early in the second half to give Bremen the lead.
Frankfurt's Japan midfielder Daichi Kamada had a first-half goal ruled out when VAR (video assistant referee) spotted Andre Silva was offside.
However, the Portugal international made amends when he buried Kamada's cross into the Bremen net for the equaliser on 65 minutes.
Schalke, second from bottom, has gone 22 league games without a win after being held to a 1-1 draw at home to Stuttgart on Saturday (AEDT).
A first-half header by Schalke's teenage defender Malick Thiaw was cancelled out by a leveller from Argentinian forward Nicolas Gonzalez for the visitors.
Schalke is edging towards the unwanted record of Tasmania Berlin, which now play in Germany's fifth tier and went 31 Bundesliga matches without victory in the 1965-1966 season.