Rossi finished runner-up and three seconds behind Marquez to take over at the top after Yamaha team mate Maverick Vinales, winner of the season's first two races, crashed out on the second lap.
Marquez had been on pole for the fifth year in a row at the Circuit of the Americas in Austin, Texas, but lost out at the start to Honda team mate and fellow-Spaniard Dani Pedrosa who made a great getaway from the second row of the grid.
With Vinales gone, his bike sliding from under him at turn 18, the two Honda riders battled each other for the lead until Marquez made a move stick on the ninth of 21 laps and began pulling away.
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Rossi was handed a 0.3 second penalty for gaining an advantage by cutting the track on lap seven but made sure that was of no consequence by passing Pedrosa for second place on lap 19 and finishing more than two seconds clear.
The Italian now has 56 points after three races with Vinales on 50 and Marquez moving up to third on 38.
(Reporting by Alan Baldwin in London, editing by Ken Ferris)
