The pair appeared evenly matched early on but the 17-year-old Gauff handed Stephens a critical break with a double fault on the last point in the ninth game and never regained the momentum from her 28-year-old opponent.
American Frances Tiafoe caused the first big upset by beating third seed Stefanos Tsitsipas 6-4, 6-4, 6-3.
Stephens overcame an attacking Karolina Muchova, ousting the Czech 18th seed 6-3, 7-5.
The Romanian will not play at Roland Garros this year because she is recovering from a calf injury.
Serena Williams survived a third-round scare to topple 2017 champion Sloane Stephens 2-6, 6-2, 6-2 and keep alive her bid for a record-equalling 24th Grand Slam title.
Former US Open champions Serena and Sloane Stephens are set to compete at the event set to begin on August 10.
Former world No.1 Clijsters and 2017 US Open champion Sloane Stephens are among the headliners as World TeamTennis kicks off its three-week regular season on Sunday, at the Greenbrier resort in West Virginia.
Konta, who had never before won a match here, will face either Petra Martic or Marketa Vondrousova for a spot in the final.
She is the first player to go through the season-ending tournament undefeated since Serena Williams in 2013.
Former US Open champion Sloane Stephens triumphed against an inconsistent Karolina Pliskova 0-6, 6-4, 6-1 in one hour and 55 minutes.