Price dropped just two legs as he outclassed Czechia's Gawlas on day six.
Despite slipping briefly in the second set, the Welshman regained control to reel off six consecutive legs and cap a 3-0 win with two 120 checkouts.
"I played pretty decent, I think, in that game, apart from the two legs I gave away," Price told Sky Sports after the match.
"I needed to improve to win this tournament, but for a first-round game, that was okay. The first round is always difficult, but to play someone like Adam, who's slow, it's hard work to get your rhythm."
Sixth seed Noppert was tested by fellow Dutchman Van der Velde, who tried to fend off the infamous Ally Pally wasps with a bottle of spray.
Van der Velde snatched the second set as the pesky pest buzzed around, but Noppert did enough to secure a 3-1 victory.
Germany's Niko Springer, meanwhile, was dealt a shock 3-1 defeat at the hands of Joe Comito, and compatriot Ricard Pietreczko beat Jose de Sousa by the same score.
Dobey through as Soutar wins epic
Chris Dobey safely negotiated a tough test earlier in the day, beating China’s Xiao Chen Zong 3-1 despite patchy finishing, the eighth seed powering through with heavy scoring to reach round two.
Scotland’s Alan Soutar survived a barely believable contest against Finnish debutant Teemu Harju, missing 15 match darts and requiring sudden death to seal a dramatic 3-2 victory.
Elsewhere, Scott Williams entertained the crowd with a swaggering 3-0 win over Paolo Nebrida, highlighted by a no-look 180 and precise finishing.
Debutant Justin Hood also impressed, averaging close to a ton and hitting five maximums in a composed 3-0 defeat of Nick Kenny.
Read our full report on Tuesday's afternoon session
Results
Afternoon session
Alan Soutar 3-2 Teemu Harju
Nick Kenny 0-3 Justin Hood
Scott Williams 3-0 Paolo Nebrida
Chris Dobey 3-1 Xiao Chen Zong
Evening session
Ricard Pietreczko 3-1 Jose de Sousa
Danny Noppert 3-1 Jurjen van der Velde
Gerwyn Price 3-0 Adam Gawlas
Niko Springer 1-3 Joe Comito
