The Gamecocks were seven-over through six holes in the fourth and final stroke-play round at Greyhawk Golf Club in Scottsdale.
But a timely eagle from Scottish star Darling at the par-5 seventh - team-mate Mathilde Claisse emulated the feat - settled the nerves.
South Carolina closed with a one-over total to finish on the same mark, setting a new programme record by nine shots.
The last-day effort saw the Gamecocks move up two spots to claim the No 4 seed, setting up a last-eight clash on Tuesday against Southern Cal.
Darling was South Carolina’s top player over the opening four days, carding rounds of 71-69-76-70 to finish joint-18th in the individual standings behind world No 1 Rose Zhang (Stanford).
Head coach Kalen Anderson. ““We have a lot of talent and kids who want to go for things and be aggressive; as they should because they are really talented.
"We are learning when to be aggressive and when not to. Over the course of the spring, they have done a really good job and bought into that.”
In the tie against Southern Cal, Anderson has handed Darling the anchor role against Amari Avery, who finished three shots behind the Broomieknowe player in the stroke-play stage.
Top-ranked Stanford earned the No 1 seed after finishing five shots ahead of Texas as Zhang, who shot rounds of 72-67-71-68 defended her individual title by a shot from San Jose State’s Lucia Lopez-Ortega, one of the players competing as an individual.