The award is in honour of three-time major winner Stewart who died in 1999 in a plane accident, after the aircraft lost cabin pressure and all on board were incapacitated due to hypoxia or a lack of oxygen.
The award is given out annually to the golfer who "best exemplifies the values of character, charity and sportsmanship."
"To win an award that has his name on it and to look at the recipients below me is an honour and it is humbling," said Furyk.
Furyk had 10 birdies, one eagle and no bogeys during Sunday's round to become the first to 58 and the only player in PGA history to break 60 on two occasions.
"I felt like someone else leaped into my body and was making the swing," Furyk said earlier of his round.
The timing of the announcement had nothing to do with Furyk's iconic performance, according to PGA commissioner Tim Finchem.
"I want to point out that this decision on his recognition of the Payne Stewart Award was made prior to him shooting 58," Finchem said.
Furyk said he would now have to find some way to get rid of most of his 59 score paraphernalia.
"I've got like 300 pictures at home with 59 on them," Furyk said. "Gotta throw all that crap away."
Furyk will receive the award on September 20.
The 46-year-old Furyk had a late start to the season due to a wrist injury that prevented him from playing in the Masters. He returned to Tour play in May and later tied for sixth at the 2016 US Open.
