Just as it seemed Hamilton was heading for his eighth world title, Verstappen took advantage of a crash by Nicholas Latifi that effectively set up a one-lap dash for the title.
Verstappen and Hamilton, who began the race level on points, started that lap wheel to wheel but the 24-year-old Red Bull driver pulled away to take the chequered flag.
It was a compelling end to a championship for the ages, played out over 22 spell-binding acts.
Hamilton had been coasting towards bettering Michael Schumacher's seven world crowns, prompting Red Bull team principal Christian Horner to say: "We need a miracle."
It came in the unlikely shape of Latifi, who crashed his Williams, with Verstappen promptly pitting for fresh tyres.
When racing resumed for the 58th and closing lap at the Yas Marina circuit Verstappen barged past Hamilton to take the chequered flag, sparking joy on the Red Bull pitwall and fury at Mercedes.
Mercedes team principal Toto Wolff lodged appeals over the result, claiming the rules governing safety car procedures had been breached.
After a lengthy hearing, the stewards rejected the protests. Mercedes announced their intention to contest the verdicts and have 96 hours to formalise or withdraw an official appeal.
Verstappen was meanwhile revelling in his new status as world champion.
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