Artjoms Rudnevs added the second just past the hour to give Hamburg, who lost their first three games of the campaign, a third win in their last four outings.
Visiting keeper Rene Adler was barely troubled as his side climbed to fourth place with 13 points from nine games.
They are 11 points adrift of leaders Bayern Munich who host Bayer Leverkusen on Sunday aiming to extend their perfect start.
Hamburg, missing Czech midfielder Petr Jiracek through injury, went ahead in the 13th minute.
Augsburg lost possession trying to play their way out of defence, Rudnevs intercepted the ball and found Son who cut inside his marker and curled a shot into the bottom left-hand corner.
Twenty-year-old Son was also involved in Hamburg's second goal in the 63rd minute as he fed Rafael van der Vaart on the edge of the area.
The Dutchman slipped a pass between two defenders, wrong-footing the Augsburg rearguard, and Latvia forward Rudnevs stole in to fire the ball into the roof of the net for his third goal since his close season move from Lech Poznan.
Augsburg, who have won only once in the Bundesliga so far this term, are third from bottom of the 18-team standings with six points from nine games.