Media playback is not supported on this device
It is the sixth time Englishman Adams, the number two seed, has lost in round one in 23 visits to Frimley Green.
Smith will next face 15th seed Brian Dawson, who came from 2-0 down to beat 2001 Lakeside champion and fellow Yorkshireman John Walton 3-2.
Dutch fifth seed Wesley Harms scraped past Japan's Seigo Asada 3-2.
Belgian eighth seed Geert de Vos, Latvia's Madars Razma and England's Mark McGeeney - who will play defending champion Scott Mitchell in round two - were the other male players to progress.
English duo Deta Hedman and Zoe Jones, seeded three and seven respectively, booked quarter-final spots in the women's draw.
The scoreline in the Canadian's favour ended up being a convincing one, but the final result could have been much different had 59-year-old Adams not wasted several chances at doubles to seize the early initiative.
In truth, Adams did not play too badly. His scoring was excellent for the most part and he threw a 10-dart leg at the start of the second set.
But 40-year-old Smith reeled off five successive legs to clinch victory, the last of them typifying the composure that has become his trademark since his Lakeside debut a year ago, when he reached the semi-finals.
Adams needed to break the throw to keep the match alive and left himself on double top after 12 darts. Up stepped Smith to land a nerveless 120 checkout and secure his place in the second round.
On this evidence, he will take some stopping.
Media playback is not supported on this device
Gary Robson is a Lakeside veteran, with 15 World Championship appearances to his name, but this year's 13th seed proved he is a bit of a novice when it comes to using a hoverboard.
The 48-year-old from Northumberland crashed into a door while trying out the gadget at his hotel a couple of days before his first-round meeting with Razma.
Robson was back to his feet again by Tuesday, but he exited the tournament with a 3-1 defeat by the unseeded Latvian.
Martin Adams: "Jeff could go on and win it. If he carries on playing like that, he's going to be in the mix at the end. From what I've seen, he's played the best darts on that stage so far."
Jeff Smith: "If I'm hitting doubles, I can compete with any player in the world and I know that. Martin's a dangerous player. If you give him an inch, he'll take a mile, so you have to finish him off."
BBC pundit Bobby George on Smith's victory: "Jeff didn't miss a shot, and he never let Martin in. He can finish, and in this game you can score all you want, but if you don't hit the double you don't win."
The second round of the men's draw gets under way on Wednesday, with world number one Glen Durrant facing Larry Butler of the USA in the day's early session (13:00 GMT, live on BBC Two).
Smith plays Dawson in the late session, while fourth seed Jamie Hughes takes on Razma for a place in the last eight.

All 54 countries at the organisation's congress in Cairo agreed to change the statutes which previously stopped officials serving past the age of 70.
It was proposed Caf should bring its rules in line with those of world football's governing body Fifa.
Hayatou, 68, will now be able to stand for another term in office in 2017.
His current mandate ends in two years and he is seeking four more years until at least 2021, when he turns 75.
The rule change follows success in recent years in adapting the statutes to limit potential opponents to Hayatou's rule.
Caf previously brought in a rule that candidates for its presidency can only come from the ranks of its own executive committee, a tight-knit club closely controlled by Hayatou. Fifa does not have the same restriction.
Cameroon-born Hayatou, a former athletics official, is the longest serving senior member in Fifa structures, where he acts as vice-president, and has had few serious challengers for power in Africa since first winning election in 1988.
Three-time champion Martin Adams was whitewashed 3-0 by Canada's Jeff Smith in the first round of the BDO World Championships at Lakeside.
[NEXT_CONCEPT]
The Confederation of African Football voted on Tuesday to remove an age limit on its officials, paving the way for Issa Hayatou to stay on as president.