The Bavarians, more used to spending the season battling relegation than dreaming of a Champions League spot, picked up where they left off before the winter break to join third placed Borussia Moenchengladbach on 30 points.
The hosts scored twice late in the first half with Halil Altintop and Tobias Werner in a game played in heavy snowfall.
But Roberto Firmino pulled one back on the stroke of halftime to keep things finely balanced.
Augsburg, who should have scored more goals in the second half, needed the skills of keeper Alex Manninger, who denied David Abraham a last-minute equaliser with a superb save, and Raul Bobadilla's stoppage time goal to confirm their 10th league win.
Argentine Franco Di Santo scored two sublime goals to lead Werder Bremen to a 2-0 home victory over Hertha Berlin in the only other game on Sunday and lift them further away from the relegation zone.
The 25-year-old forward struck just before the break, after Werder dominated but had failed to convert several good chances, curling a shot from a tight angle past goalkeeper Thomas Kraft.
He then killed off Hertha's growing momentum in the second half with a sensational volley in full flight in the 69th minute for his eighth goal of the campaign as Werder moved up to 12th on 20 points, three above the drop zone.
Lacklustre Hertha, who were missing half a dozen players and have now won one of their last six league games, dropped to 15th on 18.
Leaders Bayern Munich saw their gap cut to eight points after crashing to a 4-1 defeat at second-placed VfL Wolfsburg on Friday as the Bundesliga returned from its mid-season break.
Last season's runners-up Borussia Dortmund are in last place after a goalless draw at Bayer Leverkusen on Saturday.
(Reporting by Karolos Grohmann, editing by Mark Meadows)
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