The British pair did battle for more than two-and-a-half hours at Flushing Meadows before Broady eventually came from behind to win (4)6-7 6-3 6-4.
Broady almost found herself on the back foot right from the off when Robson earned a break point in the very first game, but the world number 82 saved it to avoid a shaky start to the match.
Robson was forced to save one herself in the fourth game, but for the remainder of the opening set she kept Broady at bay while almost making the breakthrough against the serve too.
Broady rescued four more break points to take the first set all the way to a tie-breaker, but ultimately it was Robson who came out on top 7-4 to move in front.
Broady, making her US Open debut, wasted no time in responding with a break in the opening game of the second set, only for Robson to hit right back with a break of her own to restore parity.
However, Broady finished the set by reeling off four games in a row to take it 6-3 and force a decider, before saving two break points in the first game of the third set.
The third went with serve up until the final game, when Robson's unforced error handed Broady the victory and a spot in the second round of the season's final Grand Slam.