Four-time champion Verstappen started second on the grid but took the lead from polesitter Norris just seconds into the 50-lap race en route to a flawless victory on Sunday afternoon (AEDT).
Third in the standings, the Red Bull heavyweight claimed his sixth win of the year with a monster 20.741-second buffer over McLaren's championship leader Norris.
Mercedes driver George Russell was third after fierce attempts to usurp Norris, followed by Australia's Piastri.
Russell's rookie teammate Kimi Antonelli had crossed the line fourth but dropped to fifth following a five-second penalty for a false start.
Piastri has now fallen short of the podium in his past six races, after leading the championship by a season-high 34 points following the Dutch Grand Prix in August.
Norris, while missing an opportunity to claim three straight wins from pole position, at least extends his lead to 30 points over the Aussie.
Verstappen is 42 points adrift leading into the final two rounds in Qatar and Abu Dhabi.
"It's still a big gap, but we always just try to maximise everything that we've got and this weekend, that was first," Verstappen told Sky Sports.
Norris acknowledged his mistake in the opening seconds had cost him.
The Briton had brilliantly defended against Verstappen off the starting line but went too wide on the way to turn one, with the gap also allowing compatriot Russell to sneak into second.
"I let Max have a win. I let him go, let him have a nice race," Norris joked.
"No, I just braked too late - it was my F-up.
"It was not my best performance out there, but when the guy wins by 20 seconds, it's because he has just done a better job."
Piastri was just as unlucky in a chaotic start, making contact with New Zealand's Liam Lawson before tumbling to seventh.
Racing Bull rookie Lawson had suffered heavy damage to his front right wing and finished 16th after qualifying sixth.
Piastri escaped sanction for the collision, with Lawson appearing to turn into the side of the McLaren after braking heavily to avoid Russell.
The Melburnian had been handed a controversial 10-second penalty a fortnight ago in Brazil for a three-way crash with Antonelli and Charles Leclerc, in a huge blow to his title hopes.
Sauber rookie Gabriel Bortoleto and Aston Martin driver Lance Stroll did not cross the finish line in Las Vegas after a race-ending collision on the opening lap.
Bortoleto is set to come under investigation for making contact with Stroll at the first corner.
Ferrari driver Lewis Hamilton was impressive, recovering from 19th following a disastrous qualifying performance to finish 10th.
