Payton coached the Saints from 2006 to 2021, guiding them to their first Super Bowl title in the 2009 campaign.
His Broncos handed the injury-hit Saints their fifth straight NFL defeat, a downward spiral that followed a pair of blowout victories to open the season.
"It was hard coming here because there were so many memories," Payton told the Amazon Prime post-game show. "I tried to downplay it as best I could. It's about the players, and, I mean, they played their tails off."
Denver's rookie quarterback Bo Nix connected on 16 of 26 passes for 164 yards and added 75 yards on the ground.
Wil Lutz booted four field goals and Cody Barton returned a fumble 52 yards for a touchdown as the Broncos bounced back from a loss to the Los Angeles Chargers on Sunday.
Williams grabbed his first rushing touchdown of the season when he ran untouched into the end zone for an eight-yard TD midway through the second quarter that gave the Broncos a 13-0 lead. Denver had already had field goals of 46 and 32 yards from Lutz.
Williams added a five-yard scoring run in the third quarter and Lutz added field goals of 52 and 38 yards as the Broncos dominated.
Saints rookie quarterback Spencer Rattler, making his second career start in place of injured Derek Carr, absorbed six sacks.
He lost two fumbles - one resulting in Barton's touchdown - while completing 25 of 35 passes for 172 yards before he was replaced by rookie Jake Haener in the fourth quarter.
Haener threw for 38 yards, including a 12-yard touchdown pass to Cedrick Wilson when the game was well out of hand with 1:22 remaining.
The Saints, already struggling with a hefty list of injuries, saw cornerback Paulson Adebo exit in the first half with a right knee injury.
Adebo, hurrying to provide support for teammate Jordan Howden's tackle of Williams, grabbed his leg as soon as he went down. Medical staff placed his leg in a vacuum splint and Adebo - who had a team-high six tackles before his exit - was carted off the field.
It was another blow for a Saints team that was already missing Carr with an oblique injury and tight end Taysom Hill with a rib injury.
Wide receiver Chris Olave was sidelined with a concussion and receiver Rashid Shaheed was placed on injured reserve earlier Thursday.
