Hamilton started from second on the grid but raced past his Mercedes teammate on the first corner to establish a lead he seldom looked like giving up.
Red Bull moved to within a point of second-placed Ferrari in the constructors' standings as Daniel Ricciardo secured third spot and teen Max Verstappen was fifth.
Ferrari's Sebastian Vettel was fourth while teammate Kimi Raikkonen battled up from 14th on the starting grid to take sixth on a circuit where overtaking is notoriously difficult.
Rosberg was fortunate to remain on pole after a dramatic Saturday qualifying session saw him pip his teammate at the very last after a slip-up by McLaren's Fernando Alonso disrupted Hamilton's final lap.
A stewards' inquiry later ruled Rosberg hadn't broken any rules through the course of his fastest run despite double yellow flags being waved to indicate drivers must slow.
Hamilton had been disappointed at that turn of events but all was forgotten as he took Sunday's race to Rosberg from the get-go.
"The start was everything," Hamilton later told reporters.
"I got a good start... I mean I had one of the Red Bulls on the inside of me, so I was pressured quite a lot into turn one. But the team did a fantastic job with the strategy, preparing the car as always.
"This is a great result as a team, so yeah, what a day," he added.
By comparison, the German lamented his comparative lethargy on the grid.
"It was down to the start in the end," Rosberg said. "I lost out a little bit and then into turn one, with Daniel (Ricciardo) on the outside (and) Lewis on the inside I ran out of space so I had to bail out of it."
Yet Rosberg remained keen to look forward and the potential to get back on track at next week's German Grand Prix in Hoffenheim.
"The next race is coming up very quickly, it's my home race and that's going to be awesome," Rosberg said.
Further back in the field, Fernando Alonso scored seventh position for McLaren.
Carlos Sainz Jr. of Toro Rosso, Valtteri Bottas of Williams and Force India's Nico Hulkenburg closed out the top 10.
Hamilton is now six points clear in the championship standings with 10 races remaining in the 2016 season.
Sunday's victory was also Hamilton's fifth at the Hungaroring circuit -- a record.
Prior to Sunday's race, the British driver was tied with Michael Schumacher on four.