Ferrari's Charles Leclerc overtook Red Bull's Sergio Perez on the last lap to grab second place ahead of the Mexican with French driver Esteban Ocon of Alpine finishing fourth.
Verstappen had been highly critical of the return of Formula One to Vegas after a 41-year absence but as he drove past the finish line he celebrated with a chorus of 'Viva Las Vegas' on the team radio.
The grid was packed with a host of celebrities from the world of music, sport and entertainment as memories of the embarrassment of the opening practice session on Thursday being stopped early due to a loose water valve on the track, faded.
With low grip on the track there was no shortage of incident, overtaking and lead changes as the night race delivered the entertainment the American promotors had craved from the new event.
The Dutchman won by 2.070 over Leclerc but had to work for his victory in what was the latest starting race in F1 history.
Verstappen, starting second on the grid, grabbed the lead in the first turn of the opening lap but was penalised five seconds for pushing Leclerc wide.
But Ferrari's Monegasque driver Leclerc kept in close touch and was able to get the lead back, overtaking Verstappen on lap 16.
Verstappen had to serve his penalty in the pit, leaving him plenty of work to do to regain his lead and when Leclerc went into the pits on lap 22, Perez took over at the front.
Leclerc and Perez jousted for the lead but Verstappen was ominously gaining ground and after he passed Perez he then made his decisive move on lap 37.
From then on, Verstappen never looked back, Leclerc slipping into third spot after he out-braked himself and allowed Perez to fly by.
But there was one more twist in the tale as Leclerc brilliantly moved past Perez to grab second place.