Woodhouse had knocked out defending champion Luke Littler on Monday, but Aspinall won the first set 3-0 and hit eight 180s with a 96.27 average.
Woodhouse hit a 135 checkout in the second set and took the third to a deciding leg, but 'The Asp' ultimately won it with a 106 finish on double 16.
The 16th seed has now won his last eight matches against Woodhouse.
His next opponent, Joyce, beat eighth seed Danny Noppert 3-0, winning each set 3-1, and finished the match with a 96 checkout on double eight.
Noppert, a semi-finalist last year, complained to the referee about a creak in the stage after losing the first set and never recovered.
Price sets up Clayton clash
World Cup teammates and regular rivals Gerwyn Price and Jonny Clayton will meet in an all-Welsh quarter-final after Price beat Ross Smith 3-0.
Avenging his recent defeat at the World Series of Darts Finals, 'The Iceman' won the match with an 87 checkout on double 18.
The fourth seed had been the only player to average over 100 in the first round, and Smith's 116 checkout in the third set only held him off for one more leg.
Clayton beat 12th seed Wessel Nijman 3-1 with the help of a staggering four ton-plus checkouts.
The 2021 champion won the first set 3-0 with finishes of 112, 116, and 125, the last of which included two bullseyes.
Nijman took the second set but kept missing doubles, and Clayton broke him in the third set with a 126 on the bull.
Clayton then missed three match darts, and Nijman missed double 16 for a 152 checkout before Clayton won the match on double two.
The second round concludes on Thursday as Niels Zonneveld plays Gary Anderson, James Wade takes on Stephen Bunting, Luke Humphries meets Jermaine Wattimena, and Gian van Veen faces Ryan Searle.