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Leverkusen took the lead after only 11 minutes when Stefan Kiessling found himself with space inside the box, leaving the striker with time to convert into the bottom corner.
Ronny Philp sought to reply for Augsburg as he worked his way down the right side of the box, though he was only able to crash his effort over the bar.
Andre Hahn almost found the equaliser having latched on to Philp's cross, but he sent his header past the post with the keeper scrambling.
The hosts pulled things level after the break when Tobias Werner headed into the net following Philp's telling cross into the box.
Leverkusen regained the lead through Son in the 80th minute, when the South Korean converted from a difficult angle following Julian Brandt's through-ball.
Can wrapped up the three points with seven minutes left to play when he broke away on the counter-attack before finishing into the bottom right corner of goal.