Both men were playing in the last four for the first time, but 10th seed Smith had the pedigree against debutant Aspinall and made it count to win 6-3.
Smith made a tremendous start and took a 2-0 lead, but Aspinall fought back to level at 2-2 and then again for 3-3.
But Smith was relentless in his scoring and clinched victory at the first attempt with a 121 checkout.
An overjoyed Michael Smith shares his thoughts after his 6-3 semi-final win over Nathan Aspinall... pic.twitter.com/0rvGvLP1Se
— PDC Darts (@OfficialPDC) December 30, 2018
Smith averaged 105.22, hit 17 180s – an individual record in a World Championship semi-final – and had a checkout percentage of 51.11.
An emotional Smith, who will play the winner of the match between Gary Anderson and Michael van Gerwen in Tuesday's final, said on Sky Sports: "I've never felt like that ever. I'm so happy, I don't know what to say. I just took my chances when they came.
"I nearly made a mess of myself tonight, it felt like I was going to give birth to an ostrich egg or something. I was so nervous but I'm so happy I held it together. I'm one step away from living my dream."
Aspinall had no regrets at the end of his brilliant run, saying: "I've enjoyed every single minute of it. I never thought I'd get to the semi-final.
"It wasn't my night but I'm so happy to be involved in a quality game. I wish Michael all the luck in the world because he deserves to win this."