The Iceman – whose 2021 title win came behind closed doors at Alexandra Palace – had not reached the last eight at any major tournament this year, but wasted little time in racing through the opening set without dropping a leg.
Irishman Barry, who had impressed in his first-round win over Kim Huybrechts, finally got on the board after the break, but Welshman Price continued to find his range – although he did need double one on his way to eventually closing out the second set.
Barry, 22, gave the crowd something to cheer when he took out 144 to level the third set at 1-1.
It was, though, only delaying the inevitable as 10th seed Price finished up 56 to coast through to the next round which takes place after Christmas.
“I probably wanted to win that match more than any other match in my life,” said Price said on Sky Sports.
“I wanted to enjoy Christmas, to just get away and get home. It didn’t matter how good or bad I played, I just wanted to get a win… and another tournament starts again after Christmas.”
In Monday’s afternoon session, four-time semi-finalist Wade had become the first seed to fall at Alexandra Palace as Wattimena produced an impressive display.
Wade, the 16th seed, won the first leg of the round two match – but then lost the next eight in succession as Wattimena took control.
The Dutchman hit a superb 126 finish to move within one leg of victory. Although Wade held his nerve to hit double 20 with his last dart to stay alive, the left-hander crucially missed four darts at double 10 in the next leg.
Wattimena, who reached the final of the European Championship in October, made no such mistake on double eight to seal a quickfire win.
“In the last year I have beaten big players on stage,” Wattimena said in a video with PDC media. “I can beat everyone at the moment, so I can make some history.”
In the evening session, veteran American Leonard Gates produced a vintage display to defeat Scotland’s Cameron Menzies 3-1.
Gates, 54, landed a monster 122 checkout – hitting two treble 18s – to seal his place in the second round with Menzies, partner of Fallon Sherrock, fighting back tears as his hopes slipped away.
Welshman Robert Owen secured his first win on the World Championship stage as he dispatched Niels Zonneveld 3-1 in their first-round contest.
Connor Scutt lost just a single leg as he swept past Ben Robb 3-0 with a match average of 101.92 to set up a second-round match against Australian number nine seed Damon Heta.
Elsewhere, England’s Luke Woodhouse enjoyed a 3-0 first-round victory over Lourence Ilagan.
Germany’s Kai Gotthardt overcame a broken dart as he marked his tournament debut with a 3-1 win over Scotland’s Alan Soutar.
In Monday’s opening match, Wesley Plaisier recovered from 2-1 down to beat Japan’s Ryusei Azemoto 3-2 and goes on to play former champion Peter Wright in round two.
