The 29-year-old Briton led from pole position and dominated the 61-laps race before a lengthy Safety Car interruption reduced his lead and forced him to attack again in the closing stages when he passed Sebastian Vettel to win.
It was the 29th win of Hamilton's career, his seventh this year and his second in Singapore, ending four-time champion Vettel's run of three straight wins on the south-east Asian street circuit.
He moves up to 241 points, three clear of Rosberg, with five races remaining in one of the most tense and closely-fought championships of recent years.
"I had dream this would happen last night," said Hamilton.
"Things have changed for me absolutely. I looked for a clean weekend and this was it. We always strive to get both cars 1-2 that was our goal, so things still to work on."
Rosberg turned up for the Marina Bay street circuit night race 22 points clear of Hamilton.
He eventually started from the pitlane, in an under-performing car with restricted electronic programmes available.
By lap five, he had climbed to 20th while Hamilton was controlling a lead of 2.3 seconds ahead of Vettel.
Hamilton pitted after 13 laps, but when Rosberg limped in one lap later, it was terminal for the German.
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