Mitchell hit a pair of go-ahead baskets and added a free throw to give Utah the lead for good in the final minute after they trailed by as many as 13 points earlier in the fourth quarter.
Joe Ingles added 20 points, seven rebounds and five assists for Utah. Derrick Favors chipped in 19 points and eight rebounds.
The Jazz (29-28) extended their NBA-best winning streak to 10 games.
Kyle Anderson scored 16 points while Pau Gasol scored 15 points and collected 15 rebounds for his 12th double-double to lead the Spurs. San Antonio (35-23) lost to Utah for the third time this season.
The Spurs played without leading scorer and rebounder LaMarcus Aldridge, who sat out because of a sore right knee.
The first quarter belonged to Ingles. He scored 10 of Utah's first 15 points, punctuated by a dunk off of a steal that put the Jazz ahead 15-10. That turned out to be the biggest lead for either team until late in the second quarter.
San Antonio took a four-point lead midway through the second quarter. The Spurs went up 38-34 behind an 8-0 run fueled by a pair of baskets apiece from Bryn Forbes and Murray.
Utah quickly answered with a 20-4 run to take a double digit lead before halftime. Jae Crowder fueled the spurt with back-to-back baskets that put the Jazz back in front again. Mitchell and Ingles scored baskets on five straight possessions -- highlighted by back-to-back driving layups from each player -- to finish off the spurt and give Utah a 54-42 lead.
San Antonio rallied in the third quarter. The Spurs went ahead on an 11-1 run and took a 66-61 lead on a 3-pointer from Davis Bertans.
Utah struggled to answer with another rally after converting just 4-of-22 field-goal attempts in the third quarter. San Antonio took advantage and built up an 83-70 lead early in the fourth quarter on a pull-up jumper from Bryn Forbes.
The Jazz chipped away at the lead. Ingles and Mitchell buried back-to-back 3-pointers to tie it at 87.
--Field Level Media