Ferrari's Carlos Sainz finished second, 3.882 seconds behind Verstappen, after passing Norris at the start of the last lap at Austin's Circuit of the Americas when the Briton went too deep into turn one.
"It feels a bit like old times. I'm very happy with today. If you look at the whole race, Ferrari was also very quick," said Verstappen.
"For us, finally we are racing again. Normally we are always looking back but now we can just do our own race, so we had good pace."
The victory was Verstappen's first of any sort since a sprint in Austria in June and meant the triple champion has now won all four 100km sprints so far this season with two more still to come.
Norris made a great start from fourth on the grid, seizing second place on the inside after passing Mercedes' George Russell and Ferrari's Charles Leclerc into the first corner and holding on until the 19th lap.
Leclerc was fourth, after an entertaining early scrap with Sainz, and Russell and seven-times world champion teammate Lewis Hamilton finished fifth and sixth despite a major car upgrade.
Russell complained already just after half-distance that his front left tyre was "toast".
The Haas pair of Kevin Magnussen and Nico Hulkenberg took the final points in seventh and eighth, a big boost for the US-owned team in their home race.
Red Bull's Sergio Perez was ninth and McLaren's Oscar Piastri 10th after a five-second penalty for forcing Alpine's Pierre Gasly off the track.
Their failure to score meant champions Red Bull trimmed McLaren's lead in the constructors' standings to 39 points.
The main US Grand Prix is on Sunday.