Stricker and Kelly, the defending champions, led for most of the day in the modified alternate-shot format. They had a two-stroke lead at 10 under until bogeying the par-5 17th when Stricker knocked his second shot near the lip of a greenside bunker.
Johnson and Poulter birdied No. 16 to get to 9 under.
The 63s tied the mark for low score in the modified alternate-shot format in the 10 years that the tournament has been played at Tiburon Golf Club.
"I think it was a dream start, to be honest," Poulter said.
The overall format record is a 57 by Fred Couples and Raymond Floyd in 1990 at Sherwood Country Club in California. The 12 teams in the tournament hosted by Greg Norman will play better ball Saturday, and a scramble on Sunday in the $3 million event.
Mark Calcavecchia and Jeff Overton, who grabbed the early lead with an eagle-birdie start, and Fred Funk and Kenny Perry were tied for third at 8 under. Justin Leonard and Scott Verplank were fifth at 6 under.
Members of those three teams in contention have good records in the event.
Calcavecchia, who is playing in his 17th Shootout, has won twice, and so has Perry. Leonard and Verplank lost in a playoff in 2006, and tied for second last year.
Norman and partner Matt Kuchar were last in the field after a 72.
Both of the leading teams stumbled down the stretch.
Sitting at 8-under through 11 holes, Johnson and Poulter bogeyed the par-3 12th and, despite the long-hitting Johnson, the duo parred both the par-5 14th and 17th.
"Parring the two par-5s was brutal," Johnson said. "It was no fun. No. 14 was playing pretty tough, but No. 17 was playing fairly easy. We hit four pretty good shots and we made a 5."
Stricker and Kelly were at 10 under through 15, then bogeyed the par-5 17th after Stricker knocked his second shot near the lip of a greenside bunker. Kelly was forced to play the ball directly to the left instead of straight at the pin, then barely got it out of the bunker. He missed a 10-footer for par. They missed the green at the last, but two-putted for par.
Johnson and Poulter both hit poor tee shots on the par-3 12th.
"I'm not taking the full blame on that one," Poulter joked, although he did admit he may have jinxed them when he said then they had gotten off to a good start.
Still, they're happy with where they're at and confident heading into the final two rounds.
"Our game, I think, is very good for this format," Poulter said. "Better ball [Saturday] gives us plenty of chances. I'm playing well. Dustin's hitting it long and straight, and that's dangerous."