Schwikert, who won the award in 2005, finished with a 39.6 while Postell had a 39.55.
Postell took a step on her final routine, the vault. She needed a 9.925 to tie Schwikert but scored a 9.875 instead.
The senior said afterward she wasn't disappointed, saying the team's performance was more important, but coach Greg Marsden knew the award meant a lot to his senior.
Postell finished third in the all-around as a freshman and second the last three seasons.
"How can you not be disappointed?" he said. "I think she will get over it and be fine, but you have to allow her to grieve a little bit."
Schwikert scored a 9.95 on the vault, a 9.925 on the floor, a 9.9 on the bars and a 9.825 on the balance beam. Postell finished with 9.9s on the bars and floor and scored a 9.875 on the beam.
Utah junior Kristina Baskett finished third, tied with Georgia's Katie Heenan and Tiffany Tolnay with a 39.525. Daria Bijak was eighth with a 39.375.
The Utes will be well-represented in Saturday's individual competition. Baskett qualified on everything except balance beam and Postell qualified on everything except the vault. Kyndal Robarts qualified on the vault with a 9.9. The gymnasts who qualified are first-team All-Americans. Postell is now a 20-time All-American, giving her the school record for the category.
Utah will start tonight on the balance beam, a rotation decided by a random draw. Alabama starts on vault, LSU has bye/floor, Georgia starts on the floor, Stanford has a bye/bars and Florida starts on the bars.
The Utes haven't started on the balance beam this season.
Marsden went with senior Katie Kivisto instead of sophomore Beth Rizzo as the leadoff gymnast on the floor. Kivisto had a good routine going until the last pass, when she fell and scored just 9.225.
Rizzo, who earned All-America honors last year on the floor, scored a 9.725 at regionals. Marsden said the coaches would discuss who to go with tonight in the floor lineup.
Southern Utah's Elise Wheeler competed as an individual, rotating with Georgia. She finished with an all-around score of 39.15. Wheeler scored 9.8 on the floor, 9.7 on the beam, 9.85 on the bars and 9.8 on the floor.