The championship leader will start at the front of the grid for the third consecutive race after edging out four-time world champion Sebastian Vettel and Williams' Valtteri Bottas.
A short shower threatened to derail Rosberg's charge as he skidded off at the first corner but, once the session resumed after the red flag, the German was always one step ahead of his rivals.
Vettel had shown Red Bull's pace as he moved into first place with seconds to go, only for the Mercedes driver to come roaring back and take pole by 0.486seconds.
Rosberg had played second fiddle to his team-mate Lewis Hamilton during all three practice sessions in Budapest, but for the second successive race the Briton endured terrible luck in qualifying.
Hamilton's perfect weekend came to an abrupt end five minutes into the opening session as his car suffered a fuel leak and caught fire, leaving the 2008 world champion stranded near the pit lane.
Daniel Ricciardo sealed fourth spot for Red Bull ahead of Fernando Alonso in the Ferrari, while Felipe Massa's sixth-place showing capped another positive day for the ever-improving Williams team.
Jenson Button took seventh to out-qualify McLaren team-mate Kevin Magnussen for the sixth time this season after the Dane spun into the barriers when the showers came, leaving him 10th.
Toro-Rosso's Jean-Eric Vergne was eighth, with Nico Hulkenberg one behind in his Force India.
Another former champion also suffered an early exit as Kimi Raikkonen's disastrous season continued.
The Finn has not finished higher than seventh in the opening 10 races and a miscalculation from his team saw the Ferrari driver finish 17th on the medium tyre.