The Welshman was in commanding form at the Lokhalle, peppering the treble 20 bed with authority and converting three finishes above 80 as he powered through the contest in just eight legs.
Six 180s and clinical doubling ensured Cullen rarely gained a foothold, as Price reinforced his credentials as one of the leading contenders for the title this weekend.
"I felt like I was playing well," Price said afterwards.
"The crowd were right behind me tonight. I usually bring the best out of everyone, but I'm ready for them.
"I'm fully focused on my own game, and it was another good performance.
"If I can build on that and get better, I think this tournament is mine."
Price will now meet Danny Noppert in the third round after the Dutchman eased past Ricky Evans 6-1.
Noppert capitalised on an unusually erratic display from Evans, converting four doubles from six attempts to wrap up a comfortable victory.
While Price set the tone with a commanding victory, the night's biggest shock arrived in an all-Dutch showdown as Niels Zonneveld stunned three-time champion Michael van Gerwen.
Zonneveld produced one of the finest big stage performances of his career, averaging 104.46 to claim a resounding 6-2 win despite Van Gerwen posting a formidable 107 average of his own.
The match swung heavily in Zonneveld's favour early on. The 26-year-old stormed into a 4-0 lead with relentless scoring and sharp finishing, including two ton-plus checkouts. Van Gerwen briefly threatened a recovery by claiming back-to-back legs, but Zonneveld extinguished any hopes of a comeback in style.
Successive 180s piled the pressure on before he sealed the contest with a clinical 100 finish in just 12 darts.
"What can I say? I'm so happy with this win," Zonneveld reflected.
"I'm in a position where I know I can do it, but you have to do it. Michael played really well, but I felt so comfortable and so focused tonight."
European Champion Gian van Veen also progressed comfortably, dispatching Poland's Krzysztof Ratajski 6-1 to maintain his strong European Tour form. The Dutchman's scoring consistency proved decisive as he kept Ratajski at arm's length throughout the contest.
Dutch representation remains particularly strong heading into the final day in Goettingen, with eight players from the Netherlands progressing to the last 16. Wessel Nijman averaged more than 100 in a commanding 6-2 victory over Mike De Decker, setting up a clash with Van Veen, while Richard Veenstra claimed the scalp of Germany's number one Martin Schindler.
Kevin Doets continued his impressive run after producing a remarkable 117.12 average on the opening day. The Dutchman followed that display with a composed 6-4 victory over Cam Crabtree, who had travelled from Washington DC as a late replacement for Jonny Clayton.
Dirk van Duijvenbode survived a deciding-leg thriller against two-time European Champion Rob Cross, while Jermaine Wattimena struck eight maximums in a high-scoring encounter to edge past Germany's Lukas Wenig in similar fashion.
Germany's hopes of a home success remain alive thanks to Niko Springer, who defeated world number six Stephen Bunting in another last-leg decider. Springer now faces Ross Smith for a place in the quarter-finals after the former European Champion overcame Dave Chisnall 6-4, aided by a pivotal 144 checkout midway through the match.
Damon Heta delivered a strong finish to defeat Darius Labanauskas 6-3. The Australian followed finishes of 92 and 110 with a 13-dart leg to close the match, finishing with a 101 average and his first European Tour stage victory since August.
Ryan Searle also impressed, averaging just under 100 to defeat Northern Ireland's World Cup champion Daryl Gurney 6-3 and progress to a third-round meeting with Veenstra.
Chris Dobey produced one of the most clinical displays of the session to dismantle Luke Woodhouse 6-1, earning a clash with Premier League star Josh Rock, who came through a gruelling contest against host nation qualifier Paul Krohne.
Veteran Austrian Mensur Suljovic completed the last-16 lineup with a spirited comeback victory over Karel Sedlacek, reeling off four consecutive legs from 4-2 down to set up a meeting with Van Duijvenbode.
Round Three takes place on Sunday afternoon from 11:00 GMT, before the European Darts Trophy champion is crowned later in the evening as the quarter-finals, semi-finals and final conclude proceedings in Goettingen.