Story of the race
Hamilton led Rosberg through Turn 1, with Kimi Raikkonen's Ferrari taking third from the supersoft-shod Daniel Ricciardo (Red Bull) and Vettel, as Verstappen spun at the first corner and tumbled down to last after making contact with the Force India of Nico Hulkenberg.
Hulkenberg passed Force India teammate Sergio Perez for sixth at Turn 1 at the start of lap two. In the battle between the Williams, Valtteri Bottas passed Massa for ninth on lap three.
Hamilton eased out of DRS range of Rosberg by lap five, while Ricciardo remained in Raikkonen's DRS but didn't have the top speed to make an attack.
Bottas reported: "Something is wrong in the rear damper, car is jumping like hell."
Massa repassed Bottas at Turn 14, with the Finn pitting on lap seven.
Meantime, the recovering Verstappen regained his place in the top 10 on lap seven.
Hamilton pitted to ditch his ultrasofts for softs at the start of lap eight, as did Raikkonen – which held up Hamilton's departure. Hamilton rejoined in front Verstappen.
Rosberg pitted a lap later, allowing Ricciardo to lead on his supersofts, and this time it was Vettel coming into the pits which held Rosberg's release.
Rosberg rejoined just ahead of Raikkonen, but behind Verstappen.
Ricciardo pitted at the start of lap 10, switching his supersofts for softs, and rejoined right behind Raikkonen. Ricciardo attacked at Turn 8, while Verstappen defended second hard from Rosberg.
By lap 11, the top five were covered by 5s. Rosberg was warned to not fight with Verstappen, and to look after his right-front tyre.
As the dusk descended, Hamilton was warned his tyres were losing temperature and could suffer from graining. He stayed well clear of Verstappen, however, with Rosberg clear of the battling Raikkonen and Ricciardo, with Vettel holding on in sixth.
Ricciardo launched a huge attack on Raikkonen into Turn 11, briefly getting ahead but overshooting, and allowing Vettel to get right on his tail.
Rosberg only had to finish third to clinch the championship, as the pair entered their second title showdown in three seasons.
In winning the title, he matched his father Keke's feat 34 years ago.
Hamilton tried all he could in the closing stages to back Rosberg up into the chasing Ferrari of Sebastian Vettel and Red Bull of Max Verstappen – who made a remarkable comeback from a first-corner spin – but to no avail.
In their final F1 races, Williams's Felipe Massa and McLaren's Jenson Button suffered contrasting fortunes.
Massa finished in the points in ninth, Button was forced out with a suspension failure after he hit a kerb too hard.