The pair had twice met in the final but this time they crossed paths in the second round at Alexandra Palace.
Lewis, who is now ranked 35, made the brighter start and took the first set 3-2 but Anderson powered back to prevail 3-1 by taking nine of the last 10 legs.
The Scot, who hit three maximums and averaged 93.74, told Sky Sports: "I have got a lot of respect for Aidy. He is one of the good guys.
"I'm trying a new set of darts. The doubles were all right, the scoring wasn't quite there but it's going well."
Anderson, who turns 51 on December 22, will not play again until the third round begins after the Christmas break.
He said: "I usually play two or three games before Christmas but I've played just one. I'm actually going to have my birthday at home for the first time in nine years. I'll enjoy every minute of it.
"I'm feeling good. I've just to try to play darts, (take) each game as it comes. I have not got any high hopes. I'll just throw (darts)."
Rest of the night
In the day's other second-round match, world number 22 Daryl Gurney made it through with a 3-1 victory over a battling Ricky Evans.
The former World Grand Prix and Players Championship Finals winner triumphed after Evans missed four darts to take the match to a deciding set.
In first-round action, Lisa Ashton's hopes of winning a first match at the tournament were dashed as she lost 3-0 to Dutchman Ron Meulenkamp.
Another Dutch player, Chris Landman, claimed a 3-0 win over fellow World Championship debutant Scott Mitchell while Irishman Steve Lennon edged out Latvia's Madars Razma 3-1.
Hampshire's Chas Barstow enjoyed his first appearance in the event as he beat Canada's John Norman Jnr 3-1 to set up a meeting with three-time world champion Michael van Gerwen.
Ireland's William O'Connor held off a fightback from American Danny Lauby to win 3-2 and Ryan Meikle of Ipswich beat German teenager Fabian Schmutzler 3-0.