Barnes, who also shared the lead after round two, carded his second straight 4-under 68 to tie Wilkinson in the lead at 13-under-par 203.
Cliff Kresge posted 5-under 67 for the second day in a row to move into third place at minus-12. Fran Quinn, who shared the lead after two rounds with Barnes, managed a 1-under 71. He shares fourth place with John Ellis (68) at 10-under-par 206.
The second round was completed earlier Saturday after a 65-minute fog delay Friday delayed action at Shadow Hills Country Club. That fog delay eventually caused play to be suspended for the night due to darkness.
Once the third round started Saturday morning, it became a shootout between the top three players on the leaderboard. Kresge trailed the leaders by two entering the third round.
He quickly erased that deficit as he flew into the lead with five straight birdies from the second. That got him to minus-12. He picked up another birdie on the ninth, but a double-bogey on 12 dropped him one stroke below Barnes.
Barnes was steady on the front nine with birdies on four and six. He birdied the 10th to get to 12-under and the lead thanks to Kresge's double-bogey. Barnes responded to Kresge's birdie on 13 with a birdie of his own on 11.
The 25-year-old Barnes stumbled to a bogey on 12 to slip back into a share of the lead with Kresge at minus-12.
Kresge parred out from the 14th to finish at 12 under.
As Barnes parred five holes in a row from the 13th, Wilkinson moved into the mix. The New Zealand native had collected back-to-back birdies from the fifth.
Around the turn, Wilkinson birdied 13 and 16 to grab a share of the lead at 12-under. He parred 17, but collected his fifth birdie of the day at the last to finish at 13 under. Wilkinson leads after three rounds for the first time in his career.
Barnes responded with a birdie of his own at the last to share the lead after round three. He also owns a share of the lead after three rounds for the first time in his Nationwide Tour career.
Peter Tomasulo carded a 5-under 67 to move into a tie for sixth place at 9-under-par 207. He was joined there by Chris Baryla and Chad Collins. Ryan Armour, Steve Marino and Boo Weekley are one stroke further back at minus-8.