It was Rosberg's third win of the year and sixth overall as he extended his lead in the drivers' championship over Hamilton to 29 points.
Valtteri Bottas came third for his first career Formula One podium while Williams teammate Felipe Massa, who started from pole Position, took fourth.
Four-time Formula One champion Sebastian Vettel had an engine problem in the second lap and was doubled by the field before quitting the race in the 36th on Red Bull's home circuit.
Hamilton, who was ninth after qualifying, used a blistering start and earned four places from the start and won another place to work his way up to fourth in the opening lap.
Rosberg immediately overtook Bottas but lost that position again shortly after the first turn.
Massa dropped from pole to fourth after the top-four had their first pit stops.
Sergio Perez remained in the lead until the 27th, overtaken by Rosberg who accelerated for the fastest lap so far to create distance to Hamilton, who got stuck behind Perez.
From lap 30 on, the four leaders battled it out with clear advantages for both Mercedes drivers.
Rosberg remained in the lead for the rest of the race, not allowing a pushing Hamilton a chance to overtake him.
The race once more confirmed Mercedes has best adapted to the introduction this season of the 1.6-liter turbocharged V6 engines with build-in energy recovery system.
Hamilton won four races in a row while Rosberg never finished worse than second.
Mercedes' stranglehold on victories was broken two weeks ago in Montreal by Daniel Ricciardo's maiden win, though Red Bull failed to threaten Mercedes again on its home circuit in Austria.
Mercedes dominated practice Friday with Rosberg and Hamilton posting the fastest times, but Williams hit back the next day when Bottas led the final practice before Massa grabbed pole position -- his first in six years.