The Mercedes driver won his home race for the second time in his career, and closed to just four points behind team-mate Rosberg in the standings after the German retired with a gearbox failure.
Hamilton, who has suffered two retirements already this season, said after the race that he can focus on a straight fight with Rosberg now the gap has been closed in the championship.
"I came into this weekend saying it would be great if the fans could put the wind in my sails to regain the momentum," said Hamilton.
"I got the win today and got the points back. I have been chasing all year after losing points in the first race; it has been very, very difficult.
"Now I feel we are back and close, and with the pace I had today I feel I have to focus for the next part of the season.
"Today I felt from the start I had the pace of Nico, and I was hunting him down like never before.
"I was looking forward to a wheel-to-wheel battle, but I'm sure we'll get more in the future."
Hamilton added that his costly mistake of abandoning his final lap in qualifying served as a wake-up call ahead of Sunday's race.
"Yesterday was a kick in the balls and I had to pick myself up," he said.
"I had to pull up my socks and get on with it if I wanted to win this championship.
"I cannot have situations like yesterday, in the last two races I had the pace for pole but I made it harder for myself and I will try not to do that again in the future.
"Yesterday was a really difficult day - I went away feeling terrible for the fans because I let them down, and the team, and myself.
"To turn that emptiness and negativity into a positive today was my main priority."
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