Hertha dominated the early stages with Brooks heading them into the lead from close range in the 12th minute as the visitors' unconvincing defence failed to clear their lines following a corner.
Pal Dardai's side, who had lost their previous three matches, extended their advantage in the 37th minute, Stocker tapping home after excellent interplay between captain Vedad Ibisevic and Salomon Kalou.
"We would've liked to have got that third goal in the second half," Dardai told journalists, "but we more than deserved the win today.
"Everyone did their job. It was a real team performance."
Hertha remain seven points behind Borussia Dortmund in the fourth Champions League qualifying position.
Augsburg suffered an injury blow shortly before kick off when Raul Bobadilla tweaked his calf and was replaced in the starting lineup by Alfred Finnbogason.
Lacking any real offensive threat, they remain in a relegation playoff spot on 29 points, now only one ahead of Ingolstadt after their exhilarating victory over bottom club Darmstadt.
Markus Suttner settled the relegation dogfight when he curled home a 25-metre free kick in the 72nd minute to take Ingolstadt up to 28 points and leave the visitors adrift on 15.
Pascal Gross fired Ingolstadt ahead in the 20th minute, but two goals in six first-half minutes from Darmstadt's Mario Vrancic threatened to turn the contest on its head before Almog Cohen equalised and Suttner completed the comeback.
Both sides lost a player to a red card in the dying stages as Romaine Bregerie and Antonio-Mirko Colak were dismissed after an altercation.
(Reporting by Ed Dove; Editing by Ian Chadband and Clare Fallon)
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