The world number six began the encounter at Flushing Meadows brightly and made the most of his domination to claim a break of his Belgian opponent's serve to open up a 4-2 advantage.
However, 21-year-old Goffin responded with an immediate break of his own, before going on to hold his own service game to level up the score at four games apiece.
Both players comfortably served out for a 5-5 scoreline, but then Berdych made the decisive breakthrough with a string of forehand winners to eventually take the opening set 7-5.
Berdych required just one break of serve in the next set to open up a 2-0 lead courtesy of a 6-3 success.
The third set saw the duo both hold the advantage courtesy of a succession of deuce games and break points. There was a stage were Goffin led 3-1, but the Czech Republic's Berdych eventually ran out a 6-3 victor.
He will now face Estonia's Jurgen Zopp for a place in round three.