Crane posted a nine-under 63. Both he and Wilson played the Palmer Private course at PGA West on Friday. Toms, who shared the first-round lead, posted a seven-under 65 on the Nicklaus Private course at PGA West.
The three are three strokes clear of the field at 16-under-par 128.
The field plays the two courses at PGA West as well as La Quinta Country Club over the first three days before the Palmer course hosts the final round.
Each professional is paired with an amateur during the first three rounds. Former President Bill Clinton, who lent his name to the event this year, will be among those competing in the third round.
This is the first time since 1964 that Bob Hope's name isn't associated with the tournament, though his name remains on the trophy.
Camilo Villegas, who shared the first-round lead with Toms, shot four-under 68 on the Palmer course to end 36 holes at 13-under-par 131.
He was joined in fourth place by Harris English (62), Chris Kirk (63) and Bobby Gates (63).
Chris DiMarco (64) and Brandt Snedeker (68) are among six players tied for eighth at minus-12.
Wilson got off to a hot start with birdies on his first three holes. Birdies on the sixth and eighth helped him make the turn at 11-under.
The 37-year-old birdied the 10th before converting a 15-footer for eagle on the par-five 11th. Wilson was two off the lead at that point, and he remained there with four straight pars.
Wilson moved into a share of the lead with birdies at 16 and 18.
"I just rolled in some long putts early...and that kind of got me going," said Wilson, whose 62 was a career low. "And I didn't make any bogeys and just did all the things you're supposed to do to shoot a low one."
Crane started his round on No. 10 at the Palmer course and he tripped to a bogey there. He bounced right back with an eagle on No. 11. Crane holed out for birdie from a greenside bunker on the par-three 12th.
The four-time winner on the PGA Tour dropped in back-to-back birdies at 14 and 15. Crane tripped to a bogey at 16, but birdied his next two holes to head to the front side at minus-12.
Crane birdied the second, fourth and sixth to climb up the leaderboard. He finished at 16-under after making a 17-foot birdie effort on the ninth.
"I made a lot of 10- and 15-footers and I made all my kind of close putts for birdie," Crane stated. "I had 22 putts today, so I just putted extremely well and every part of my game felt great. I guess that's how you do it."
Toms birdied the 10th on the Nicklaus course, but gave that right back as he bogeyed the 11th. He carded birdies at the 14th and 16th to get to 11-under.
The 2001 PGA Champion parred five in a row around the turn. Toms ended that run with a birdie on the fourth. Over the last three holes, he picked up four strokes.
Toms birdied the seventh and ninth to go with an eagle on the par-five eighth.
"Tee shots were a little erratic today, I guess you would say. I really didn't do much of anything," Toms admitted. "I started to hit some better shots, hit some good putts that didn't go in, and then made a few and then finished great. I turned just a so-so day into a pretty solid day."
Ryan Moore had the round of the day. Playing on the Nicklaus course, Moore bogeyed the second and third. He erased those mistakes with an eagle on the par-five fourth.
Moore poured in 11 birdies in a 12-hole span from the sixth to soar into a share of 14th at 11-under-par 133. He birdied six in a row from the sixth, then five straight from the 13th.
NOTES: Dustin Johnson and Scott Stallings both withdrew on Friday...Johnson, who is recovering from knee surgery, withdrew due to a stiff back, while Stallings had a rib injury...Villegas in the only non-American among the top 13 on the leaderboard.