Seventh seed Wade is the winner of 11 major titles but has never cracked it at Alexandra Palace, falling in the semi-finals four times over his stellar career.
'The Machine' laid down a marker against the energetic Azemoto, though, dropping just one leg throughout the match en route to a comfortable victory.
"I'm really happy," Wade told Sky Sports. "I haven't won a game here for three years! Tonight showed the practice I've been doing.
"It wasn't spectacular, but when I needed to hit something and do something, it happened.
"I have felt better coming here, but have also felt a lot worse. I have got two wonderful children, a wonderful wife, I'm 42 years old, I'm old! I'm just enjoying things."
Wade and Azemoto exchanged breaks in the first set, with the Englishman claiming one more after an 87 checkout to nudge into the lead.
He broke instantly in set two by taking out 100 and swept the remaining couple of legs for 2-0, despite needing seven darts to nail double 10.
Wade ramped up his scoring power in the third and broke Azemoto's throw with a well-executed 85 checkout via tops.
The 42-year-old completed a comprehensive outing by removing 76 on double eight to set up a second-round clash with Ricky Evans.
Van Barneveld dumped out
Van Barneveld won just two legs in a 3-0 drubbing against Stefan Bellmont, who dazzled to become to first Swiss player to ever win a World Championship match.
Legendary Dutchman 'Barney' reached the fourth round two years ago before succumbing to a debuting Luke Littler, so he had plenty of prize money to defend.
But it wasn't the start he was looking for as Bellmont grabbed a break of throw then held to claim the first set, ending three for three on the doubles.
The Swiss doubled his lead by opening with a 96 checkout and capitalised on three missed Van Barneveld darts at tops to break, before sweeping the frame to make it six straight legs won.
Bellmont showed no sign of slowing down and added another two more in set three until Van Barneveld finally stopped the rot with an 80 checkout, with the legs streak ending at eight.
A dejected-looking Van Barneveld survived three match darts in the fourth set but failed to take out 142, with Bellmont wrapping up a convincing win on double four.
American debutant Adam Sevada won the opening match of the session by defeating Canada's top player Matt Campbell 3-1.
Arizona native Sevada won each of the last five legs to come out on top in the all-North American clash.
Wednesday's PDC World Darts Championship results
Raymond van Barneveld 0-3 Stefan Bellmont
Martin Schindler v Stephen Burton
