Nico Rosberg placed second despite having to pit early in the race but now trails his teammate by 17 points in the drivers’ standings, with three races remaining.
Hamilton won his fourth straight race and his ninth of the season and was presented the winner’s trophy by Russia’s president Vladimir Putin, who had pushed for his country to have a Formula One race.
“We’ve had an amazing week, Russia’s been so good to me,” Hamilton said from the podium.
The Briton lost the pole when Rosberg cut a corner and pushed past at the start of the race, but Rosberg braked hard in the second turn and had to come in for a set of new tires.
They lasted him until the end as he fought his way from the back of the pack.
Hamilton took the flag 13.6 seconds ahead of Rosberg. Valtteri Bottas was third in a Williams, 17.4 seconds back.
Hamilton became only the fourth driver in Formula One history to win nine races in a season.
He equalled Nigel Mansell’s record of 31 victories for a British driver.
The ninth 1-2 for Mercedes this season is one short of McLaren’s 1988 record.
There are 100 points at stake in the last three races of the season - in the United States, Brazil and the double-pointer in Abu Dhabi.