Shameka Christon, Danielle Adams and Danielle Robinson each had 15 points for San Antonio, which has lost five of six.
DeWanna Bonner had 14 points and Candice Dupree had 12 rebounds for Phoenix. Diana Taurasi, who came in averaging a league-leading 24.6 points, was limited to 18 points for the Mercury, but Griner more than made up for it.
San Antonio gave away green shirts to commemorate its "Go Green" recycling campaign, but the attire might as well have been passed out in tribute to Griner.
The former Baylor standout attracted the largest crowd of the season, which included Danny Green and Boris Diaw of the San Antonio Spurs.
Griner, who starred for four years in nearby Waco, was 11 for 19 from the field while showing off an expanding offensive game with a series of spins and bank shots to overcome San Antonio's early physical defense.
The teams exchanged baskets in the fourth after each made huge runs in the previous quarters.
San Antonio opened the second half on a 20-4 run to take a 53-47 lead with 3 minutes left in the third. Adams hit a pair of 3s and Robinson had six points in the run.
After Griner banked in a jumper in with 8:26 left in the third, Phoenix failed to score until Bonner hit a pair of free throws with 2:50 remaining in the quarter.
Despite the offensive futility, Phoenix only trailed 56-53 entering the fourth after a stellar second quarter.
Griner was unable to gain positioning early against the burly Adams.
She was 0 for 2 with no rebounds in the opening 4 minutes of the game before making a layup with 5:12 left in the first. Her lone dunk attempt was stuffed by the rim, but she gathered the miss and put it back up for a layup.
Adams proved more effective, scoring nine points in the first quarter while going 3 for 4 from the field.