McCall swatted Macy Miller's shot in the key with 1 second left to seal it. Thompson, who missed a 3-point try with 18 seconds to go, made a driving layup with 8.2 seconds left and converted the free throw as the Cardinal erased a five-point deficit over the final 3 minutes.
Briana Roberson hit a 3-pointer with 1:11 left that pulled Stanford within 64-63.
What a finish for Tara VanDerveer's team in the 1,000th game at Stanford for the Hall of Fame coach.
The Cardinal (26-7) will face Notre Dame on Friday in the Lexington Regional.
South Dakota State (27-7) missed a chance at reaching the Sweet 16 for the first time in program history despite pushing the Cardinal to the very end at Maples Pavilion where they have been so dominant in the NCAAs.
Immediately after Thompson's missed 3, Miller couldn't convert the second of two free throws. VanDerveer called timeout before Thompson's winning play.
Stanford survived for its 14th straight NCAA win at home since Florida State stunned the Cardinal 68-61 at Maples in the second round on March 19, 2007.
The Cardinal overcame shooting 10 for 22 from the free-throw line, while the Jackrabbits were 6 of 16 after also struggling in their first-round upset of fifth-seeded Miami on Saturday.
Miller converted a three-point play with 6:01 left to put South Dakota State ahead 58-51, then Ellie Thompson hit from long range at 4:41 for a 62-54 lead that forced Stanford to use a timeout.
McCall scored 13 of her 20 points in the first half but was largely held in check the rest of the way. Thompson wound up with 19 points and five assists.
Gabby Boever was held to three points on 1-of-7 shooting in her final collegiate game, while Thompson had 14 points and Kerri Young 13.
South Dakota State's loyal, blue-clad traveling fan base is headed home from rainy Northern California to snow in the forecast.
The Jackrabbits reached the second round for the first time in 2009 after five first-round exits.
South Dakota State: After struggling taking care of the ball late in a first-round win against No. 5 Miami, the Jackrabbits didn't the ball over once in the opening quarter and finished with seven turnovers. ... The Jackrabbits have never reached a regional in school history. ... They had won five straight games and 18 of 21. ... There will still be more NCAA women's basketball in basketball-crazed South Dakota, with Sioux Falls hosting a regional this weekend.
Stanford: VanDerveer, who coached her 1,203rd career game overall, WOULD be 21 victories from joining Pat Summitt as the only NCAA women's coaches with 1,000 career wins. ... The Cardinal improved to 34-4 in NCAA games on their home floor. ... The 65 NCAA women's games played at Maples are second most at any venue behind Tennessee's Thompson-Boling Arena (66).
Stanford: Faces Notre Dame on Friday.