The brothers, who say they are innocent, are accused of having formed an organised criminal network in early 2021 in Romania, as well as in the US and Britain
Simpson nabbed his first title since the 2018 Players Championship, firing a two-under par final round of 69 on the TPC Scottsdale Stadium Course for a 17-under total of 267.
Trailing Finau by one to start the day, Simpson was two down through 16 but two-putted from 52 feet for a birdie at the par-five 17th and calmly rolled in a 17-footer for birdie at the 72nd hole.
Finau still had an eight-foot birdie look at the last hole of regulation that would have won it, but he couldn't get it to drop and the tournament went to a playoff for the fourth time in five years.
Once again, Simpson put his tee shot in the fairway, then drained a 10-foot birdie putt as Finau, who was in a bunker off the tee, was left searching for an elusive second PGA Tour title to go with his Puerto Rico Open victory in 2016.
"It was a lot today," said Simpson, the 2012 US Open champion. "Tony is a great champion. He played awesome today. To come down to the last couple of holes and then birdie 18 twice feels great."
Finau had bounced back from a bogey at the eighth with an 18-foot birdie at the 12th and added another birdie at the 13th despite missing the fairway.
But the American, who fired a nine-under 62 on Saturday to take the 54-hole, wouldn't find another birdie and settled for a one-under par 70.
"Hats off to Webb," Finau said. "A couple birdies coming home and if you're going to get beat that's how it should happen.
"I definitely didn't give him the tournament. He took the lead from me early and I got it late, but unfortunately it's how the cookie crumbles today."
Simpson's chances of stealing the victory looked slim after he found the water at the 15th and bogeyed to fall two adrift.
Neither player could birdie the par-three 16th, the hole lined by stands holding thousands of raucous fans.
The pressure on Finau increased when he found a fairway bunker off the tee at 17 and Simpson bore down for the win.
"Honestly there on 15 when I drove it in the water and Tony hit a great drive - I didn't think it was over but I thought I was going to have to do something special," Simpson said.
For Finau, who was a late entry to the tournament after initially committing to the Saudi International, it was a disappointing first Phoenix Open since his recent move to Scottsdale.
"It's a bitter week as I look back on it," he said. "But man, lots of stuff to be learned.
"I had a chance, a great chance to win this week. Unfortunately, it didn't happen, but my game's better than it's ever been."
Final scores
267 - x-Webb Simpson 71-63-64-69, Tony Finau 69-66-62-70,
270 - Justin Thomas 68-68-69-65, Bubba Watson 69-66-69-66, Nate Lashley 66-67-69-68
271 - Max Homa 72-67-64-68, Scott Piercy 67-65-68-71
272 - Adam Long 66-68-66-72
273 - Daniel Berger 69-71-66-67, Billy Horschel 63-68-73-69, Branden Grace (RSA) 67-67-70-69, Jon Rahm (ESP) 67-68-68-70, Mark Hubbard 69-68-64-72, An Byeong-hun (KOR) 65-66-70-72, Hudson Swafford 66-67-66-74
The brothers, who say they are innocent, are accused of having formed an organised criminal network in early 2021 in Romania, as well as in the US and Britain
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