Verstappen had started the race in 18th after being handed a double penalty for a suspension failure and needing a new gearbox.
But he made an excellent start in keeping his nose clean to avoid the chaos on the opening lap, moving him swiftly up to ninth on the grid.
And the Dutchman was soon in a battle for the podium as he raced his way through the pack.
While most of his rivals were on two-stop strategies, Verstappen had to make do with one, meaning he was nursing his tyres towards the end of the race with Hamilton hot on his tails.
The final 10 laps saw the front three of Kimi Raikkonen, Verstappen and Hamilton locked in a fascinating battle.
Hamilton, desperate to clinch his fifth world title in Austin, tried desperately to overtake Verstappen and made his move with a few laps remaining.
However, the Briton yielded somewhat when he realised there was no room to pass his rival.
The duel between the Mercedes and Red Bull drivers was praised by many for the respect shown between the pair.
But former British Formula One driver Brundle expects more of a stranglehold the next time the two competitors meet on track.
“In the closing stages it became clear that Kimi, Max and Lewis would be on the podium, but we had no idea in which order,” Brundle wrote in his Sky Sports column.
“You can never have too much Lewis v Max, and the young Dutchman won this encounter with clever placement of his car, and not the late change of direction tactics he used in his early F1 days.
“He’s brilliantly established himself in the psyche of others to be approached with caution. Just as Ayrton Senna did.
“I’m surprised Lewis admitted in the press conference that he’d given Max too much space, it only underlined the hierarchy on this matter.
“We may well see some flying carbon fibre as drivers try to mark their territory in future.”
Brundle went on to insist he hopes to see Hamilton win the world title in style at the Mexico Grand Prix.
“At least that keeps the championship alive for another seven days, with Lewis only needing seventh or better to take his fifth title,” he added.
“I hope he takes the championship with at least a podium, it’s rubbish when you crown a champion as he loiters in the garage or runs back to the pits as in Mexico last year.”