The 21-year-old, who switched her allegiance from Australia to Great Britain earlier this year, got off to a positive start at Flushing Meadows.
Having held all of her service games, the 21-year-old went on to take the opening set 6-1.
Hungary's Babos appeared to have secured the second set when she broke Konta's serve for a 5-3 lead.
However, Konta broke immediately to reduce the arrears to 5-4, before going on to emerge victorious from the next three games to seal her slot in the next round, where she will now face either Olga Govortsova or Tamira Paszek,
Today's victory was her first ever at Grand Slam level.