Davies missed six darts on the double in the opening set and mistakenly attempted to leave the stage at 2-2, as Littler eased to a straight-sets win.
The teenager will now face Mensur Suljovic in the last 32. "I watched (him) this afternoon, and he played well," Littler told the BBC.
"When the tournament draw came out, Mensur messaged me and said: 'See you in round three...' Mensur, let's have a good game."
Ninth-seeded Price became the 12th seed to exit after losing to Plaisier, despite posting the higher average of 95.83 to 94.28.
"I can't believe it, I'm so over the moon. It's my biggest victory ever. I don't know how I did it," said the Dutchman after his win.
Germany's Martin Schindler, meanwhile, defeated Keane Barry in his round-two tie, as Australian Damon Heta overcame Stefan Bellmont of Switzerland.
Cross through as Cullen frustrated
Earlier, former world champion Rob Cross advanced with a composed 3-1 victory over Ian White, sealing his progress with a decisive 82 checkout in a final-leg shootout.
White briefly rallied to level after a second-set whitewash, but Cross regained control with a clinical 113 finish and never looked back.
Joe Cullen, however, exited in bad temper after a 3-1 defeat to Suljovic, the 32nd seed, visibly irritated by his opponent’s pace of play and engaging in a frosty exchange after.
Elsewhere, Luke Woodhouse swept aside Max Hopp 3-0 with a 107.36 opening-set average, while Krzysztof Ratajski overcame Ryan Joyce 3-1.
Read our full report on Sunday's afternoon session
Results
Afternoon session
Ryan Joyce 1-3 Krzysztof Ratajski
Joe Cullen 1-3 Mensur Suljovic
Luke Woodhouse 3-0 Max Hopp
Rob Cross 3-1 Ian White
Evening session
Martin Schindler 3-0 Keane Barry
Wessel Plaisier 3-0 Gerwyn Price
Luke Littler 3-0 David Davies
Damon Heta 3-2 Stefan Bellmont