The Gabon international, who now has 13 goals in 10 matches and is joint top of the scorers' list with Robert Lewandowski of Bayern Munich, grabbed two goals late in the game after also putting them in the lead in the 18th minute.
Marco Reus, continuing his revival after an injury-hit season start, struck twice, including two minutes after Aubameyang's first goal to take his tally to five goals.
Augsburg, in last place on five points, briefly cut the deficit early in the second half with their first chance of the game when Raul Bobadilla headed in a corner but could do nothing to stop Shinji Kagawa, a whirlwind in midfield, who provided three assists in total.
Aubameyang, who also scored a hat-trick in the Europa League at FK Qabala in midweek, restored their three-goal lead in the 85th, slipping in to turn an Adrian Ramos shot into the net and then chipped in his third goal in stoppage time for a hat-trick.
"I made a bet with my family yesterday that I would score three goals," the striker said. "It's great it worked out but much more important is the fact we got the three points and Marco is back as well."
"He scored two and I am so happy. He is such an important player for us."
Bayern are on a maximum 30 points after crushing Cologne 4-0 on Saturday for their record-extending 10th win in 10 league games with Dortmund in second place on 23.
Borussia Moenchengladbach beat third-placed Schalke 3-1 to make it five wins in a row under coach Andre Schubert and continue their explosive rise up the table to seventh place on 15 points, four behind their opponents.
Gladbach, who had lost their first five matches of the season before the departure of coach Lucien Favre, took the lead when Lars Stindl's penalty was saved by Ralf Faehrmann but the midfielder beat defender Benedikt Hoewedes to head in.
Schalke levelled on the stroke of halftime when Gladbach's Andreas Christensen intercepted a Max Meyer cutback only to send into in for an own goal.
The visitors stepped on the gas after the break and came agonisingly close with attempts by Leroy Sane and Pierre-Emile Hojbjerg before Gladbach struck again with Raffael's outstanding free kick floating over the wall and past Faehrmann in the 70th.
Schalke were then left with 10 men when Johannes Geis was sent off with a straight red card for a studs-up challenge on Andre Hahn, who had to be carried off and Gladbach struck once more through Julian Korb to protect their perfect record under Schubert.
(Reporting by Karolos Grohmann, editing by Pritha Sarkar and Clare Lovell)
