The Englishman was cruising at four up after seven and was three up with five to play but DeChambeau piled on the pressure as he won the 14th and 15th to take it up the 18th at Austin Country Club.
Fleetwood found a penalty area off the 18th but luckily it was dry and he was able to play out to over the back of the green and get up and down in sensational style to take a one up victory and win Group Five.
"At the end of the day the result is what matters, advancing is what matters," said Fleetwood. "Take all the positives from that.
"I got up early and I was playing really solid and then it turned really scrappy on the back nine and Bryson played more solid on the back nine.
"On 18, bit of a roller coaster, but just to give myself a chance and have the chance and hit a great putt like that, it's nice to have done it, I'd have just rather it been over earlier."
Fleetwood won the first hole with a par and the fifth and sixth with birdies before a concession on the seventh put him in control.
But he was unable to get up and down from a bunker on the next and put two balls in the water on the 11th before hitting straight back with a gain on the 12th.
DeChambeau birdied the 14th from 18 feet and Fleetwood three putted the 15th to set up the big finish.
The 2010 champion went one up as Griffin bogeyed the fourth and he holed a 25 footer on the fifth before dropping a shot of his own on the seventh.
Griffin conceded the eighth and bogeyed the ninth to put Poulter in command and while the American won the 12th and 13th, he conceded the 15th.
A birdie on the 16th saw him hit back but Poulter holed a 16 footer on the next to seal victory.
No one is going to give you a match in this format. You've got the best 64 players in the world playing against each other and it's not easy - Ian Poulter
"My game feels really good at the minute," Poulter said. "I feel very comfortable. I feel comfortable on the green. I feel comfortable if I'm missing the odd green with my short game. I'm having plenty of chances. I wouldn't say it's the best I've felt but I feel pretty good.
"You have to play well, you have to hole out, you have to hang tough when need be. It's just no one is going to give you a match in this format. You've got the best 64 players in the world playing against each other and it's not easy."
Rahm faced Ryan Palmer in a winner takes all match in Group Three but the duo could not be separated over 18 holes, with Palmer holing a 14 footer to win the final regulation hole in a match where the gap was never more than one.
Palmer never saw the fairway on the first play-off hole but got up and down brilliantly to save par and a half, before Rahm put his approach to seven feet on the second to become the only top seed to advance from the group stages.
"It was a hard-fought battle," he said. "I don't think either of us had our best. He was making putts and I was just being pretty much solid off the tee and into the green. I was just not making anything."
Van Rooyen was involved in a tight match against Brandon Todd in Group 14, taking the lead with a birdie on the third but losing it on the tenth, before Todd bogeyed the 12th and Van Rooyen birdied the 16th in a 2&1 victory.
That set up a play-off against Daniel Berger and it was a bogey on the second extra hole that handed Van Rooyen the win.
"I think the confidence will just kind of stay where it is," he said. "It's relatively high always.
"I'm happy to be playing well. I haven't sort of gotten off to the best of starts here on the PGA Tour this year, so I guess it's a good time to play well, now more than ever."