The American has not won a PGA Tour event in more than 12 years on tour, having made more money than any player without a victory with more than US dollars 11.9 million (£7.65million) in career earnings.
However, Baird has propelled himself into pole position at CordeValle Golf Club and sits two shots ahead of former leader Paul Casey, of England, and South Africa's Ernie Els.
Also notable on Saturday were the performances of Jim Renner and Will Mackenzie who set a course record with their rounds of 62, propelling both men 58 places up the leaderboard into a four-way tie for ninth.
Tiger Woods shot his second consecutive 68 and sits nine shots behind in a tie for 38th.
Els had appeared on course to head into the final round atop the leaderboard midway through the third.
However, after turning with an eagle on the ninth, Els suffered a bogey and just one more birdie on the back nine to finish with a 67 while Casey hit three birdies for 68.
Baird took full advantage with seven birdies and an eagle although it could have been even better given he bogeyed the final hole to move to 13 under overall.
Adam Hadwin, Charlie Wi, Bryce Molder and Bud Cauley are in a four-way tie for fourth, a shot further back.
As Baird stands on the brink of a first tournament win, it was suggested he could emulate Kevin Na who went 210 starts without victory before finally triumphing in Las Vegas last week.
Baird, though, told pgatour.com: "I don't look at it like if he can do it, I can do it.
"There's been guys that have played out here for a while that suddenly win a golf tournament. It's going to be - my world is not going to come to an end if I play well tomorrow [Sunday] and win the golf tournament.
"I think it's more of a bucket list type thing.
"I used to say it's not going to bother me.
"But I think looking back on it, it probably would bother me if I never did win on the PGA Tour.
"Feeling like you can win on the PGA Tour and winning on the PGA Tour are as different as black and white."
Els has 18 Tour wins and said: "This is what I've been working toward, especially after the PGA.
"I changed a lot of things and I feel that the work I've put in is starting to come through."
Casey has just one career win on the Tour, in 2009, though he has won twice elsewhere in the world this season, most recently in South Korea last week.
"I take a lot of confidence out of last week," said Casey, who has been plagued by a foot injury all year.
"Not just the win but I felt that the game is coming back to where I want it to be.
"I know it was the Korean Golf Tour, it wasn't the PGA Tour but a win is a win."