World number one Price endured no such difficulties, storming through to the last 16 in quick-fire fashion, dispatching five-time World Champion Van Barneveld in straight sets.
Price was without a win in his previous six televised tussles against the Dutchman, but he made a blistering start, converting clinical 84, 68, 87, 117 and 66 combinations to race into a two-set lead.
Van Barneveld responded with 125 and 147 finishes in his bid to launch a fightback, but he paid the price for spurning one set dart in the third, as Price sealed the deal with a superb 128 finish on double ten.
“I needed to start well, put Raymond under pressure and I did that,” reflected Price, who suffered two defeats against Van Barneveld at last month’s Grand Slam of Darts.
“In the Grand Slam I let Raymond off so many times and I didn’t want to do that tonight. I wanted to be clinical.
“I’m defending a lot of ranking points and I want to lift the trophy, so it’s been really difficult, but I’m looking forward to the last 16 and there’s not as much pressure on me now.”
Elsewhere, Gabriel Clemens survived a match dart to defeat Jim Williams in one of the games of the tournament so far, in a contest played out with over 700 German fans in attendance at Ally Pally.
The opening two sets were shared before Clemens followed up a sensational 157 finish with a 13-darter to lead 2-1, although the German number one then squandered two darts for a 3-1 lead as Williams levelled.
The Welshman – who dumped out James Wade in round two – raced into a 3-2 lead, only for Clemens to save his skin in dramatic fashion, taking out 64 and 110 finishes to stay alive.
Williams then missed one match dart at double six for a 126 show-stopper, and he was suitably punished late on, as Clemens followed up an 11-dart hold with an 85 finish on the bull to triumph in a seven-set classic.
Josh Rock moved through to the last 16 on his World Championship debut, edging out two-time semi-finalist Nathan Aspinall in a seven-set thriller to set up a showdown with Jonny Clayton.
The pair fired in 25 maximums between them in a gripping tussle, but Aspinall was punished for missing 37 darts at double as the Northern Irishman held his nerve to progress.
Both players posted ton-topping averages in a high-quality opening set, although it was Rock who drew first blood courtesy of a sublime 141 checkout.
Aspinall responded by winning six of the next eight legs to move 2-1 ahead, but Rock rallied in some style, posting six perfect darts in the latter stages of set four to restore parity with an 11-dart hold.
The 21-year-old then came through a tense fifth-set decider to regain the initiative, and while Aspinall wrapped up set six without reply, Rock followed up a brilliant tops-tops 99 finish with a two-dart 70 kill to complete a landmark win.
Rock will now take on five-time TV title winner Clayton for a place in the last eight, after the Welshman came through a gruelling five-set battle against former quarter-finalist Brendan Dolan.
After all five sets went to deciding-legs, Clayton’s superiority in the crucial moments proved decisive, with Dolan missing 157 and 85 combinations in sets two and three.
The Northern Irishman opened his account in a tightly-contested fourth set, but a clinical 116 combination from Clayton propelled him to a 4-1 success with a 93 average and six 180s.
Elsewhere, Dimitri Van den Bergh was the first player to book their place in the last 16, courtesy of an emphatic 4-1 success against Poland’s number one Krzysztof Ratajski.
Scroll down for full tournament results and a complete daily schedule
Tuesday December 27
Afternoon Session (12.30pm GMT)
Third Round (Best of 7 sets)
TV Coverage: Sky Sports Darts
Evening Session (7pm GMT)
Third Round (Best of 7 sets)
TV Coverage: Sky Sports Darts