Ferrari's Vettel, who started on pole, needs to score 16 points more than his Mercedes rival to keep the title fight open to the next round in Brazil.
That means the German has to finish in the top two, a seemingly impossible task with Red Bull's Dutch 20-year-old Max Verstappen leading into the first corner and 43 seconds ahead of Vettel after 10 laps.
Hamilton's team mate Valtteri Bottas was in second place with Force India's Esteban Ocon third.
Verstappen seized the lead with an aggressive move into the first corner, bumping wheels with Vettel, with Hamilton also trying to pass Vettel.
The Ferrari's front wing hit Hamilton's rear right tyre, puncturing it.
"Did he hit me deliberately," asked Hamilton over the team radio. "Not sure, Lewis," was the reply.
Vettel pitted for a new wing while Hamilton, who is 66 points clear of his rival with 50 to be won after Mexico, also came in for a change of tyres while mechanics inspected the car for any further damage.
Stewards decided no further investigation of the incident was necessary.
(Reporting by Alan Baldwin, editing by Pritha Sarkar)
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