Anderson moved a step closer to lifting the Sid Waddell Trophy for a third successive year after triumphing in an match of extraordinary quality, which included 33 180s - just one short of the record set in his final victory against Adrian Lewis last year.
Chisnall equalled the previous record for the most 180s hit by a player as he fired in 21 maximums - equalling van Barneveld's record from the 2007 World Championship final - but was still unable to stop Anderson charging into the semi-finals.
The number two seed edged the first two sets - which were littered with 14 maximums - before Chisnall hit back to level.
The St Helens ace, appearing in his first Alexandra Palace quarter-final, went on to lead the seventh set 2-0 with the match poised at three sets apiece, but Anderson stormed back to take a 5-3 victory.
The run extends Anderson's winning run to a record 16 matches on the Alexandra Palace stage - surpassing the previous 15-game winning streaks of Taylor from 2009-2011 and Adrian Lewis from 2011-2013 - as he continued his bid to become only the third player in the sport's history to win three successive World Championships.
"It's the kind of match this brilliant crowd deserved to see," said Anderson, who averaged 106 and produced finishes of 124, 121 and 104 in setting up a last four clash with Wright.
"I pinched that seventh set and the rest is history. I'm still here, still fighting and I'm happy with how I've been playing.
"I'm looking forward to playing Peter and I hope it will be just as good as this game was."
Number three seed Wright is one match away from a second final appearance - having been runner-up to Michael van Gerwen in 2014 - after sealing a 5-3 success against Wade, which he finished with a brilliant 134 checkout.
The Scottish ace also completed finishes of 130, 121 and 100 as he averaged over 100 for the fourth consecutive match, this time reaching almost 105, as he fended off a battling display from Wade, who at one stage took out back-to-back finishes of 144 and 155 to lead 2-1 in sets.
Wright recovered from that double blow to win three successive sets as he pulled clear to complete a memorable win.
"I'm quite happy with how I played," said Wright, who hit ten of the tie's 22 180s on route to another stunning triumph.
"I had to take the big 134 shot at the end because James is always right behind you. He is a top player and I'm pleased to get over this one.
"I'll be up for the semi-final and if I keep playing like this, you never know what could happen."
World number one van Gerwen fired in two 170 finishes in another dominant display as he saw off Northern Ireland's Daryl Gurney in straight sets to continue his bid to win a second World Championship title.
The Dutch sensation averaged 103.08 and - like Wright - has averaged over 100 in all four games in reaching the semi-finals, having won four straight sets from one-all to complete a comfortable win.
"I don't have anything to celebrate yet," said van Gerwen, the 4/7 odds-on favourite with sponsors William Hill to win the £1.65 million tournament.
"I've come here to do a job and I'm only a step closer. I knew Daryl was the underdog today and sometimes that is hard position to be in.
"This World Championship is something special. I still get goose-bumps on the walk-on and I'm excited and ready for the semi-finals."
Van Gerwen will meet his Dutch compatriot Van Barneveld in the last four after the five-time World Champion defeated Taylor for the first time in the World Championship since his famous victory in the 2007 final.
Van Barneveld produced back-to-back finishes of 167 and 120 in set four as he opened up a 3-1 lead, only for Taylor to fight back and level by taking the next two sets.
The Dutchman took the seventh set in three legs though and wrestled his way back from 2-1 down in the eighth to end Taylor's hopes of a 17th World Championship title for another year.
"I'm very proud of what I've done here so far," said Van Barneveld, who's New Year's Day meeting with Van Gerwen will see him contest his third consecutive Alexandra Palace semi-final.
"I didn't feel very comfortable at times but I was expecting more from Phil. I put him under pressure though, especially in the last set.
"I'm going to play the greatest player in the world next when I play Michael and I'm really excited. I can't wait to play him."
Van Barneveld famously knocked Van Gerwen out of last year's tournament in a thrilling last 16 clash and the pair will meet again on Sunday after Anderson and Wright face off, with the final being contested on Monday.
Van Gerwen's position as the odds-on 4/7 favourite with sponsors William Hill is followed by reigning champion Gary Anderson as the 11/4 second favourite, with Wright now 8/1 to claim his first televised title and Van Barneveld the 10/1 outsider.
Van Gerwen is 1/6 on to claim victory in the semi-final against Van Barneveld - who is a 4/1 chance of repeating his success of last year - while Anderson is 2/5 ahead of his clash with Wright (2/1).
William Hill World Darts Championship
Friday December 30
James Wade (6) 3-5 Peter Wright (3)
Gary Anderson (2) 5-3 Dave Chisnall (7)
Michael van Gerwen (1) 5-1 Daryl Gurney (24)
Raymond van Barneveld (12) 5-3 Phil Taylor (4)
Quarter-Finals - Best of nine sets
Sunday January 1 (7pm)
Gary Anderson (2) v Peter Wright (3)
Michael van Gerwen (1) v Raymond van Barneveld (12)
Semi-Finals - Best of 11 sets