"This was a huge game for us," BYU coach Jennifer Rockwood said. "It was a battle with lots of emotion. Our fans create such an awesome atmosphere here and I'm so excited for Kayla, she really came through for us tonight by hitting that penalty kick to win the game."
Marquette scored the first penalty kick but after two amazing saves from BYU goalkeeper Erica Owens and the game-winning penalty kick from Varner, BYU reached the Elite Eight for the first time since 2003.
Owens made seven saves to earn the shutout, while Marquette's Amanda Engel made nine saves.
BYU will face either North Carolina or Baylor in a quarterfinal game Friday.
The BYU men's cross country team finished sixth at the NCAA Championships on Saturday in Louisville, Ky.
Jared Ward led the Cougars with a 14th-place individual finish in 29 minutes, 33.4 seconds on the 10K course, while BYU's Tylor Thatcher (29:58.0) finished 34th, Conner Peloquin (30:36.7) was 84th, and Thomas Gruenwald (30:39.0) was 91st.
Oklahoma State won the team title with 72 points, followed by Wisconsin (135), Colorado (158), Northern Arizona (191), Florida State (238) and BYU (245).
Kennedy Kithuka of Texas Tech won the individual title in 28:31.3.
Southern Utah's Nate Jewkes (30:22.8) was 54th, and Utah Valley's Josh McCabe (30:39.8) was 94th.
In the women's 6K race, Utah's Amanda Mergaert finished 14th in 19:57.7.
Weber State had its best-ever team finish, at 17th, led by Amber Henry (20:11.6), who placed 32nd, and Sarah Callister (20:38.3), who placed 79th.
Oregon (114) won the women's team title, followed by Providence (183) and Stanford (198).
Iowa State's Betsy Saina (19:27.9) won the individual title.
San Diego 3, BYU 2 • In San Diego, the Cougars clinched a share of the West Coast Conference title despite losing to the Toreros, 25-23, 28-26, 13-25, 23-25, 15-3. BYU (25-3, 13-3) tied with San Diego in the standings and split two matches with the Toreros, but received the league's automatic bid to the NCAA Tournament due to a 5-4 edge in sets in the season series.
Arizona 3, Utah 1 • In Tucson, Ariz., Utah's Chelsey Schofield had 10 kills, but the Utes lost the Pac-12 match 25-20, 21-25, 25-17, 25-18. Alli Spurrier tied her career high with nine kills for the Utes (13-18, 4-18).