Price came out of the starting blocks perfectly and took the first set. Barry then fumbled on a double 16, which gave Price the chance to make it 2-0.
The Irishman missed a 164 finish to break back and then had to watch as Price checked 112 to make it 3-0.
Both players had incredible problems with doubles in the second set; Price missed 10 times and Barry 17.
In the end, it was the experienced Welshman who secured the win.
Shortly before the end of the match, Barry secured a second leg and even showed that he could hit doubles after all with a 144 checkout.
Price ultimately reached the third round rapidly after a rock-solid performance.
His opponent will be either Joe Cullen, Wessel Nijman or Cameron Carolissen.
Owen to face Clemens
In the first match of the evening, Gabriel Clemens was particularly keenly watched.
Thanks to Robert Owen's 3-1 win over Niels Zonneveld, the German number two - who may be number three after the World Championship - knows his opponent for the second round.
Owen lost the first set without a leg of his own, but then completely turned on the heat and won the next three sets.
In the end, he was the luckier player in a relatively even match.
Then, it became clear when Connor Scutt defeated Ben Robb 3-0.
It was almost a complete whitewash, but the Kiwi managed to secure a single leg in the second set.
Scutt dominated with an average of almost 102 points and hit over 50% of his doubles.
So, there will be no Oceanic duel in the second round - but a match between the Englishman and Australia's Damon Heta.
Gates stuns Menzies
Cameron Menzies had a disastrous evening as he was defeated 3-1 by Leonard Gates
The fan-favourite Scotsman had extreme problems with his game.
A total of 27 double attempts missed their target and 'Cammy' burst into tears during the final leg of the match, not least because fans cheered every time he missed a dart.
For Gates, there will be a second-round duel with Nathan Aspinall.