Verstappen had been all set for a procession to a third race win of the season after cruising past polesitter Leclerc within 10 laps of the race start.
But a late safety car when Pierre Gasly took a wheel off the McLaren of Lando Norris and sent the Briton spinning paved the way for a 10-lap shoot-out between the two championship rivals in a race which was in danger of being a damp squib .
In the end, the straightline speed of the Red Bull meant Verstappen was able to keep the chasing Leclerc at bay to close the gap in the championship standings to 19 points and take the win by nearly two seconds.
An error in qualifying had left Verstappen third for the start in the heat of Miami but he eased past a slow-starting Carlos Sainz to move into second.
Leclerc had pulled clear in the race lead but his Dutch rival increasingly closed the gap and had passed both Ferraris for the lead within 10 laps.
A first race in Miami had brought out the A-listers onto the grid, among them the David Beckham, the Williams sisters and Pharrell Williams. But the danger was the second American race on the calendar would fail to live up to the celebrity hype.
In Formula One’s grand new era of overhauled regulations, there were no shortage of overtaking maneuvers for those fighting for the minor positions.
But for F1 managing director Ross Brawn and his team, the worry is that up front there was little real overtaking opportunity until the field got bunched up with Norris’ misdemeanor.
The only chance of turning the race into any sort of spectacle rested on rain which had threatened to fall again as it had pre-race or else a crash. In the end, Gasly provided that as, clearly unsighted and damaged from a tangle with Fernando Alonso, he clipped Norris and sent him spinning at high speed.
It brought out the virtual safety car and then the safety car itself and, when racing restarted, Leclerc, who had earlier in the race complained about the Ferrari being difficult to drive, threatened to get past but couldn’t quite make it stick.
There was a similar Ferrari-Red Bull fight between Sainz and Sergio Perez for third but that went the Ferrari’s way.
George Russell, who struggled to make the hard tires work and dropped well back at the start, again produced another drive of damage limitation to finish in the top five for a fifth straight this weekend. Lewis Hamilton was a place behind in sixth.
For Mercedes, it was a weekend that finally delivered some promise although its car still blows hot and cold. There is confidence of an improvement in Spain in two weekends’ time.
But heading to the Circuit de Catalunya, the danger is that Red Bull have marked themselves out as clearly the fastest car if not undone by reliability issues with Ferrari increasingly struggling to get close with any consistency.