Penny Taylor had 19 points for first-place Phoenix (9-3), which led by 15 in the second half before having to hold off the Shock. Tulsa pulled within six in the final 7 minutes.
Griner was coming off a big night on Wednesday in a win over Minnesota in which she set a career-high with 27 points and tied a career best with 18 rebounds. The second-year standout topped her personal scoring mark against the Shock, shooting 12 for 14 from the field to go with four rebounds and six blocks.
She said she didn't feel like she was in a special rhythm or anything on this night.
''If it goes down into me and they double(-team), kick out and if they don't, hey,'' she said, explaining her night offensively.
Phoenix coach Sandy Brondello knows her budding star has begun to make her mark.
''This is what she does, she's a game-changer,'' Brondello said. ''In the middle, she anticipates shot blocking so well.
''She's playing great. We need everyone gunning but she has that dominant presence inside it just helps our team function on all levels.''
Tulsa coach Fred Williams agreed.
''For us, she's gotten a lot better getting her balance and getting a second-effort shot so we dropped two players on her and she was still able to spin and make some moves,'' Williams said.
Glory Johnson had 20 points and 11 rebounds for the Shock (4-7), who lost back-to-back nights to Los Angeles and Phoenix after winning four straight.
Tulsa center Courtney Paris added 15 points and 15 rebounds. The Shock stayed with the Mercury even after falling behind, outrebounding Phoenix 42-29 but shot just 6 for 24 from the field in the final quarter. They were also 2 for 18 from beyond the arc.
''They are a lot bigger than we are and they have the size and are shot blockers,'' said Johnson, who has three straight double-doubles. ''We try to work within and not change our game or change our shooting but they altered our shots and it was tough for us.''
Phoenix improved to 5-1 at home and kept Tulsa the league's only team winless on the road at 0-6.
Skylar Diggins, fifth in the league in scoring at 18.9 points per game, was held to 12 on 4-of-15 shooting. She had seven assists.
The teams split their first two meetings this season, each having won at home.
Taylor, who has fought injuries in the latter part of her 13 years in the league, had 10 points in the first quarter and said she is playing pain-free.
''Well, it's the Penny Taylor of old,'' Brondello said. ''It's vintage Penny Taylor. I mean, that's what she does. She's getting her legs under her basically, she's feeling good. Any day she could do that, so I was really happy for her.''
For Taurasi it was her third straight double-double and fourth in her last five games.
''You've got Brittney's presence inside, and she was great, 12 for 15, that's exceptional, but it can open up from outside. It's more about knocking them down when you are open,'' Brondello said. ''It just kind of opens up the whole floor. They are not sure who to guard. They are going to trap (Griner) and leave someone else open and they have to pick their poison.''
The Mercury led by as much as 11 in the first half but Tulsa closed to 45-38 at the break, led by Johnson's 12 points.
Sparked by Taurasi's four-point play, the Mercury ran off a 16-4 spurt in the third quarter to open up a 65-50 lead with 3:30 left.
However, Tulsa bounced back with its inside game as Paris got a couple of easy putbacks to pull the Shock to 66-59 going into the final period. The margin could have been closer but Jennifer Lacy missed three of four free throws in the final minute of the quarter.