Defending champion Price had hit a nine dart finish in the second leg of set four, which was a third of the tournament overall and a record in any one World Championship – but Smith then took out 130 to level the match at 2-2.
The Iceman kept his cool to secure the fifth set with an impressive 108 finish and move ahead once again, only for Smith to respond once more as he claimed the sixth 3-0.
Smith, runner up to Michael Van Gerwen in 2019, looked to have the momentum, only to miss a chance to close out the seventh, which Price did on double top to move within a set of victory.
As tension mounted, Price paused mid throw during the fourth leg of the eighth set to ask for a heckler to be removed. Once play resumed, the Welshman had two match darts, but failed to land either as Smith forced a decider.
An early break of throw and then a 126 check-out on the bullseye put ‘The Bully Boy’ on the brink before he clinched the set 3-1 to deliver a hard-earned victory over the world number one.
Smith hit 16 180s with an average of just over 101 and speaking to Sky Sports following his win, the ninth seed said: “I just kept saying to myself ‘just focus on everything you have been working on over the last 12 months’.
“I felt like I was in control and knew if I could produce it in the last set, then I would be OK then.”
On the heckling of Price, who is often the target of the boo-boys, Smith said: “Someone at the back kept shouting my name, and it was putting me off as well.
“As it was getting to a tighter game, Gezzy (Price) started hearing it more and more, but he has proved now for the last few years that he loves being the pantomime villain.”
In the opening match of Saturday’s evening session at Alexandra Palace, former champion Peter Wright ended the run of debutant Callan Rydz as he fought back from 3-1 down to progress on a final set tie-break.
Rydz, 23, had put out world number 10 Nathan Aspinall earlier in the tournament – and looked on course for another upset following a nerveless opening of quality finishing.
However, Wright, champion in 2020, showed all of his big match experience to twice level the contest when looking on the brink of defeat before edging the tie-break deciding set 4-2.
‘Snakebite’ hit 17 maximums and averaged 99.75, but felt he had not been at his usual clinical best.
“I am happy with the win, but throughout the match at crucial times I’d miss three darts at a double to win legs or sets, which let Callan back in,” Wright said on Sky Sports.
“I should have won it ages ago, but that was down to myself missing. But what a game of darts.”
Earlier, Gary Anderson kept alive his dreams of a third world title after beating Luke Humphries 5-2 to reach the last four for the seventh time.
Scotland’s Anderson, champion in both 2015 and 2016, averaged 96, hit eight 180s and produced four ton finishes – including a 148 checkout.
Anderson, who has struggled with a back injury for much of the year, will play compatriot and good friend Wright for a shot at a second successive appearance in the final.
“The crowd have got their favourites, and of course especially in England, they want to see the Jocks get beaten, but it actually kind of pushes me on a bit,” Anderson said to PDC media.
James Wade is in the semi-final for the first time since 2013 after a 5-0 whitewash of Mervyn King, and next faces Smith in Sunday’s showdown.
The 38-year-old is an outsider after seeing his ranking plummet in recent weeks, but ‘The Machine’ insists he is enjoying proving people wrong – especially pundit Wayne Mardle.
“I don’t mean this in a personal or derogatory way to Wayne Mardle, but he always tips me off, to not do this and not do that,” Wade said in his post-match press conference.
“I have won more than Wayne did in two weeks than he did in his entire career, but he is hell-bent on being a little bit silly.”