Alcinas opened up a 2-0 lead in the first set but a sublime 167 finish helped Wright force a decider, in which he left himself on 28 by scoring 171 – only to see his opponent take out 109.
Alcinas won the second set 3-1 before Wright seemed to start to find his feet, taking the third set 3-0, but the Spanish World Cup star produced a brilliant 13-darter to break in the deciding leg of set four to complete the biggest shock of the tournament so far.
“I played really well and I’m happy,” said Alcinas, after defeating Wright for the first time in his career, having lost seven previous encounters against the Scotsman.
“I had played against Peter several times and could never beat him. He didn’t play his best game but it was still a hard match and I had to play well to win.
“It is a special moment for me and I hope I can go even further now.”
Dave Chisnall survived a scare as he fought back from two sets behind to defeat Josh Payne and reach round three.
Payne, a winner on the PDC ProTour in 2018, looked to be cruising to victory when he took the first two sets without dropping a leg while averaging almost 110.
Chisnall produced a strong leg to break in the deciding leg in set three before taking the fourth set 3-1, forcing a decider, which he won by the same margin after Payne had missed a dart to level.
“I thought I was out when he was two-nil up,” admitted Chisnall.
“I just had to believe that he wouldn’t be able to keep playing as well as he was and I know that I get better the longer games go on.
“That’s what happened and it gives me confidence for the longer format in the next round.”
Luke Humphries impressed in his 3-0 win against Adam Hunt, in which he won nine consecutive legs after losing the first.
Newbury ace Humphries, who topped the PDC Unicorn Development Tour Order of Merit for the second successive year, needed less than 20 minutes to complete his first World Championship victory.
“The first set was a bit scrappy but I’m really pleased with how I played after that,” said the 23-year-old, who averaged 97.58.
“I feel like I am a better player this year and the Development and Challenge Tours have really helped me. Without them, I wouldn’t be here playing like I did.”
Former World Grand Prix finalist Brendan Dolan eased into the second round with a 3-0 win over China’s Yuanjun Liu.
Dolan won the first two sets without allowing Liu any darts at doubles before a fabulous 141 finish saw the Northern Irishman take a seventh successive leg.
“I think my experience pulled me through,” said Dolan, who is making his 11th consecutive World Championship appearance.
“It was all new to him and I made sure I started well and didn’t let him in. I’ve been practising hard this year and I feel like it is starting to pay off.”
Liu landed double 16 to get his only leg on the board before Dolan claimed the next two legs to wrap up a comfortable win.
Darius Labanauskas recovered from losing the first set to defeat Matthew Edgar on his Alexandra Palace debut.
Labanauskas, the first Lithuanian to compete in the World Championship, put 15 missed darts at doubles in set one behind him to take the second set 3-0 before winning the next two sets to set up a second-round tie with Raymond Van Barneveld.
Germany’s Gabriel Clemens was Sunday’s first winner after he eased to a straight-sets victory over Aden Kirk.
Clemens will meet Scottish ace John Henderson in round two Friday.
Irishman William O’Connor moved into the second round at Ally Pally for the first time with a 3-0 win over Yordi Meeuwisse.
O’Connor took the first set 3-1 despite a brilliant 133 checkout from his Dutch opponent, before going on to win sets two and three in last-leg deciders to set up a last 64 meeting with James Wilson.
Ross Smith became the fifth player of the day to reach round two for the first time after defeating veteran star Paul Lim 3-1.
The action continues on Monday evening with three further first round ties, including Anastasia Dobromyslova’s return to Ally Pally, before Van Barneveld’s second round meeting with Labanauskas.
The £2.5 million tournament is being broadcast live on the dedicated Sky Sports Darts channel from December 13-January 1 and worldwide through the PDC’s international broadcast partners.
2018/2019 William Hill World Darts Championship
Schedule of Play
Sunday, December 16
Gabriel Clemens 3-0 Aden Kirk (R1)
William O’Connor 3-0 Yordi Meeuwisse (R1)
Brendan Dolan 3-0 Yuanjun Liu (R1)
Dave Chisnall 3-2 Josh Payne (R2)
Luke Humphries 3-0 Adam Hunt (R1)
Darius Labanauskas 3-1 Matthew Edgar (R1)
Ross Smith 3-1 Paul Lim (R1)
Toni Alcinas 3-1 Peter Wright (R2)
Monday, December 17 (1900)
Vincent van der Voort v Lourence Ilagan (R1)
Wayne Jones v Devon Petersen (R1)
Ryan Joyce v Anastasia Dobromyslova (R1)
Raymond van Barneveld v Edgar/Labanauskas (R2)
Best of five sets. No tie-break in the first round.