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The Cook Islands claimed an early lead when Brad Takairangi picked up a high kick before pouncing for the line, but efforts from Paulo and Bureta Faraimo ensured that the Americans were level at the break.
Paulo added his second at the start of the second half, but a strong spell from the Cook Islands resulted in tries for Lulia Lulia and Dominque Peyroux, which levelled the scores with less than 20 minutes to play.
The USA edged back in from when Paulo passed the ball out side for Mark Offerdahl, who found a gap in their opponents' defence to add the fifth try of the night for America, before a late effort from Craig Priestly sealed the result.
The USA face Wales in their next match in the tournament on Sunday, while the Cook Islands play Tonga on Tuesday.