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The five-times world champion now has 79 career wins, 12 short of Michael Schumacher's all-time record with 13 races remaining this season.
The win around the streets of capital Baku was the fifth of Bottas’ career and he became the first driver this season to win from pole.
The win was the 75th of Hamilton's career, second in a row after his lucky triumph in Bahrain two weeks ago, and sixth in China.
Hamilton is now 17 wins short of Michael Schumacher's all-time record of 91.
The team, which finished eighth out of 10 in the constructors' championship last year, will remain under the same independent ownership and management with Frenchman Frederic Vasseur as principal.
The win was the 72nd of Hamilton's stellar career.
Hamilton had needed only five points to be sure of the title while four times champion Vettel had to win to have any hope of denying the Briton.
The 39-year-old Finn's victory for Ferrari in Sunday's US Grand Prix ended a winless run of 113 races dating back to victory with Lotus in Australia in March 2013.
The Briton stands on the brink of becoming only the third man to win at least five titles, with his team's stranglehold on both championships tighter than ever after he won Sunday's Japanese Grand Prix.
Ferrari's Sebastian Vettel, the Briton's closest rival, finished only sixth after an early collision with Red Bull's Max Verstappen left the German fighting back from 19th place in a damaged car.