Jan-Ingwer Callsen-Bracker scored the only goal of the game in the eighth minute, while Hamburg's woes were compounded by a red card shown to Maximilian Beister late on.
The home side had chances to level and even to win the game, but lacked incision in front of goal, while Augsburg goalkeeper Mohamed Amsif played his part in what could prove to be a crucial win for the Bavarians.
Augsburg recently won in northern Germany, beating Hamburg's local rivals Werder Bremen 1-0, and they got off to a fine start on Saturday afternoon.
Callsen-Bracker scored his first goal in just over a year, powerfully heading in Tobias Werner's free-kick in the eighth minute to silence the HSV-Arena.
Hamburg responded well to the early setback and had chances to level, with Milan Badelj wasting the first when he failed to beat Amsif when one-on-one with the keeper in the 23rd minute.
Five minutes later, Artjoms Rudnevs was just inches away from making contact with Marcell Jansen's low cross with the goal at his mercy.
Rudnevs and Son Heung-min were both then denied by an increasingly busy Amsif, who saw Rudnevs miss the target with a header just before the interval.
Son saw a goal disallowed for a tight offside decision on the hour mark as Hamburg continued to attack in the second half.
But Amsif continued to thwart the northern Germany side, whose frustration grew by the minute and culminated in substitute Beister receiving his marching orders in the 80th minute for a dangerous tackle on Daniel Baier.
Even with 10 men, Hamburg had the better chances, but Son wasted their last one as Augsburg held on to three precious points.