The German had led from the start in his Red Bull Racing car, but was passed during the second round of pit-stops by Button.
The 24 year old was then leapfrogged by Alonso at the final round of tyre stops, but needing only a solitary point to defend the title he won in Abu Dhabi last November, he cruised home in third place to spark celebrations on the Red Bull pit wall and he was mobbed by his mechanics when he returned to the pits.
Vettel has dominated the season, winning nine races and taking 12 pole positions, and he succeeds Alonso as the youngest driver to win two world titles.
His Red Bull teammate Mark Webber was fourth, with Lewis Hamilton's McLaren fifth, despite colliding with Felipe Massa's Ferrari for the second successive race.
The victory was Button's third of the season and his first in a completely dry race since the Turkish Grand Prix in June 2009.
There are still four rounds of the 2011 season still to be held, with the teams back in action in Korea next weekend.