The feud began at the PGA Championship in May and expanded over social media, with taunts of "Brooksy" being shouted to DeChambeau at tournaments.
Rahm birdied his last three holes Friday for a 5-under 65. Cantlay birdied his last two holes for a bogey-free 66 to keep one shot ahead.
At 13-under, Cantlay was two clear of Rahm, who holed out from the fairway for birdie at the first on the way to a five-under-par 65 that put him at 11-under.
The victory in the second of three PGA Tour playoff events sends Cantlay into the concluding Tour Championship atop the playoff standings and in pole position to claim the FedEx Cup top prize of $15 million.
The PGA announced that big-hitting DeChambeau was being replaced by Patrick Reed and hours later the Spanish Olympic Committee said Rahm had pulled out after testing positive for Covid-19 for the second time in two months.
Oosthuizen, looking as calm as the conditions, was at 11-under 129, breaking the 36-hole Open record first set by Nick Faldo in 1992 at Muirfield and matched by Brandt Snedeker in 2012 at Royal Lytham & St. Annes.
Spieth was a shot off the lead held by Oosthuizen, who saved par from a fairway bunker on No 18 for a 6-under 64.
One shot behind and running out of holes, Rahm made a sweeping, left-to-right birdie putt from 25 feet on the 17th hole to tie for the lead, and closed with another bending 18-foot birdie putt that gave him a one-shot victory over hard-luck Louis Oosthuizen.
Oosthuizen, the 2010 British Open champion, fired a one-under par 70 to stand on five-under 208 after 54 holes at Torrey Pines with Hughes, who fired a 68, and Henley, who saved par from a bunker at 18 to shoot 71.
US Open Golf 2021: Richard Bland, Russell Henley share lead with Louis Oosthuizen, Matthew Wolff one back
Bland had a 4-under 67 and walked off Torrey Pines with his name atop the leaderboard in only his fourth major. His first one was 23 years ago.
Both Koepka and DeChambeau said the US Golf Association never contacted them about pairing them together.
Rahm and second-ranked Justin Thomas could overtake Dustin Johnson for world number one with a victory if Johnson finishes worse than 18th.
Koepka, who shared second in last month's PGA Championship at Kiawah Island, returns to South Carolina for this week's PGA Palmetto Championship at Congaree, which replaced the Canadian Open that was canceled due to the COVID-19 pandemic.
From the tee or from the fairway, Bryson DeChambeau would hit a shot only to hear someone in the crowd scream out, "Let's go, Brooksy!"
Mickelson looked like the Mickelson of old on another windswept grind around the Ocean Course at Kiawah Island, running off five birdies over his last eight holes, the last one giving him a 3-under 69 and a share of the lead.
The resort's Ocean Course will be the longest venue in major golf history at 7,876 yards, 135 more than the mark set by Erin Hills when it hosted the 2017 US Open.
Four-time major winner McIlroy, a 32-year-old from Northern Ireland, and three-time major champion Spieth, a 27-year-old Texan, are both playing well and chasing history.
It was on the 1,000-mile flight from Charlotte to Dallas that DeChambeau got the news from his agent he was in 68th place
Struggling to make the cut to the low 50 and ties after opening 76s were reigning US Open champion Bryson DeChambeau and four-time major winner Rory McIlroy, who struggled to his worst opening round at the Masters.
Five months after Dustin Johnson shattered the course record with a 20-under par 268 in rain-softened conditons, the Masters is back in its typical April setting and serving up lightning-fast greens with firm fairways to roll balls into danger.