Verstappen appeared on course to sail to the chequered flag and emulate Michael Schumacher and Sebastian Vettel by taking a record-equalling 13 victories in a single year.
But the two-time world champion hit trouble at the Circuit of Americas with his final pit stop when he was left stationary for 11.1 seconds as Red Bull struggled to bolt on his front-left tyre.
With only 20 laps remaining, Verstappen's miserable stop allowed Hamilton to take the lead.
"Beautiful, f****** beautiful," said Verstappen on the radio as the red mist descended over the Red Bull driver.
Verstappen fell behind Charles Leclerc after his tyre trouble, but he fought his way past the Ferrari driver with 15 laps left to set up a blockbuster finish as Hollywood star Brad Pitt watched the drama unfold from the back of Hamilton's Mercedes garage.
As the lap counter ticked down, Hamilton's lead began to evaporate and with 10 laps remaining, Verstappen moved to within two seconds of his Mercedes rival.
By lap 50 of 56, the margin stood at only half-a-second as Hamilton's silver mirrors suddenly became occupied with Verstappen's blue machine. Verstappen did not waste any time to make his move.
On the 190mph charge to Turn 12, the newly crowned double world champion darted into Hamilton's slipstream before jinking to the left and placing his Red Bull on the centre of the apex.
Hamilton tried to come back at the man who so controversially denied him a record world crown in Abu Dhabi 10 months ago, but Verstappen held his nerve - despite a number of warnings for exceeding track limits - to take the spoils and deny Hamilton a first win of the season.
Leclerc finished third ahead of Sergio Perez as Red Bull secured their first constructors' title since 2013.
George Russell finished fifth, one place ahead of Lando Norris. Fernando Alonso crossed the line in seventh despite being sent airborne and out of control after he collided with Lance Stroll at 180mph.
Alonso grazed the wall before he was able to limp back to the pits for repairs. Bits of Aston Martin fell off Stroll's machine and into the oncoming traffic but the grid escaped without injury.
Alonso was later demoted from seventh to 15th after being penalised by the stewards for racing on in a damaged car.
Verstappen secured his second title in Japan a fortnight ago, but a cloud hung over his championship party in Austin following Red Bull's breach of the sport's financial rules.
On Saturday, it was also confirmed that Dietrich Mateschitz, the Austrian billionaire who founded Red Bull in 1984 and secured their entry on the F1 grid 21 years later, had lost his long battle with cancer.
An emotional Verstappen dedicated his career to Mateschitz after news broke of his death on Saturday night. And moments after, the grid stood shoulder-to-shoulder to pay tribute to Mateschitz with a round of applause, the Dutch driver delivered a textbook start to move past pole-sitter Carlos Sainz.
There would be further misery for Sainz when Russell sent him out of the race after he thudded into his Ferrari.
Verstappen was on cruise control to the flag but recovered from his desperate pit stop to claim the 33rd win of his career, moving him ahead of Alonso and into sixth on the all-time list, with only Hamilton (103), Schumacher (91), Sebastian Vettel (53), Alain Prost (51) and Ayrton Senna (41) ahead of him.
Verstappen said: "It was a tough race. It was all looking good but the pit stop was longer than we would have liked.
"I had to give it everything and I pushed to the limit to come back. This win is definitely dedicated to Dietrich and what he has done for everyone."
Remarkably, Verstappen celebrated his 25th birthday just last month.
Verstappen has failed to win only six of the 19 rounds of an utterly one-sided campaign and with three rounds in Mexico, Brazil and Abu Dhabi still remaining, it seems unconceivable he will not become the sole holder of the most wins in a single campaign.
"So close, but yet so far," said Hamilton on the radio. "Let's keep pushing."
Hamilton added: "We were so so close. I did everything I could to stay ahead but they were a little bit too quick today.
"I am shattered. The car was a handful. It felt amazing to be in the lead. That is something we have been working so hard on, and I held so much hope, but we will hold on to that, keep pushing and try to give it everything in the next three races. It will come to us."