Bayern captain Manuel Neuer was back in goal after missing three games with a thumb injury but he conceded after just 13 seconds, Leon Goretzka turning a Philipp Max cross into his own net.
Joshua Kimmich's 10th Bundesliga assist of the season helped Bayern get back on level terms, with Coman's smart low volley providing the finishing touch.
But Augsburg retook the lead in the 23rd minute as Max raided down the left wing again, his cross taking a couple of deflections on its way through to Ji Dong-won, who smashed home.
Robert Lewandowski hit the crossbar with a header from another Kimmich cross but it was Coman who levelled again on the stroke of half-time, drilling through Gregor Kobel's legs.
And within seven minutes of the restart Bayern led, Coman adding an assist to his brace by setting up Alaba for a thumping finish that beat Kobel at his near post.
Reece Oxford missed a great headed chance from yet another Max centre and Kobel denied Coman his hat-trick before a late injury scare for the winger, but Bayern held on for three vital points in the increasingly close title race.
What does it mean? Pressure piling on Dortmund
With the leaders not in action until Monday's trip to Nurnberg - and having thrown away a three-goal lead to draw at home to Hoffenheim in their last league game before again conceding three in a heavy Champions League loss at Tottenham - Dortmund are being hunted by Bayern.
Coman ready to fill winger void
Bayern's extensive targeting of Callum Hudson-Odoi during the January transfer window suggests they are concerned about the impending departures of Franck Ribery and Arjen Robben.
But a wonderful performance from the fit-again Coman, who scored his first Bundesliga goals of the season after missing three months with an ankle injury earlier in the campaign, showed they already have a ready-made replacement for one of their veteran wingers.
However, Coman needing treatment on his left ankle in the closing stages will be a concern.
Kobel positioning at fault
Augsburg goalkeeper Kobel will perhaps feel he should have done better with all three Bayern goals as he often seemed to be caught in the wrong place.
What's next?
Bayern go to Liverpool for the first leg of their last-16 Champions League tie on Tuesday before returning to Bundesliga action on Saturday at home to Hertha Berlin. Augsburg are at Freiburg on the same day.
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