Anderson hit seven 180s in a convincing win against Northern Ireland’s Burness, who had seen off Paul Nicholson earlier in the day.
Burness kicked off with a 12 darter to break the Scotsman, who edged the first set 3-2, before losing set two 3-1 despite a brilliant 130 finish on the bull.
Anderson refused to become embroiled in an early battle, however, and reeled off six straight legs to seal the success.
“It wasn’t my best performance but it was enough to get through,” said Anderson, who has won the UK Open, World Matchplay and Champions League in 2018.
“I knew he could be a tough opponent and he proved that from the start but luckily the 180s started going in and I got over the line without any dramas.”
Burness had landed five 180s and dropped just four legs in his earlier 3-0 triumph over Nicholson in round one.
Lewis moved into the third round after fending off a fightback from Cody Harris, who had defeated Martin Schindler in the first round on Thursday.
The Welshman, who reached the last four 12 months ago, raced into a two-set lead but was pegged back to two apiece, before hitting a 180 in three consecutive legs in the decider to progress.
“I started off well, went two sets up and felt like I was cruising but then things started to go wrong for me,” said Lewis, who fired in nine 180s.
“I was just happy to get over the line in the end. I just need to be more consistent now and there’s no reason I can’t go on another big run here.”
Dutchman Danny Noppert enjoyed a comfortable debut victory, reeling off nine consecutive legs to defeat Hong Kong’s, Royden Lam.
A recent semi-finalist at the Players Championship Finals, Noppert only dropped the first leg before racing into the second round, where he will meet Max Hopp on Saturday.
“I didn’t feel much pressure in that game and I’m really glad about that,” said Noppert, a winner on the PDC ProTour in 2018.
“It’s a crazy atmosphere here. I got goosebumps on the stage an really enjoyed it. It makes me feel confident and not nervous at all so we will see what happens in the next round.”
Chris Dobey also secured a 3-0 victory as he moved into the second round at the expense of Russia’s Boris Koltsov.
Two-time TV quarter-finalist Dobey dropped just two legs in an impressive opening display.
“I felt much more relaxed than last year,” said Dobey, who was beaten by 16-time World Champion Phil Taylor in the first round 12 months ago.
“When you’ve played the best in his last ever World Championship, you’re not going to get nervous again and I think that helped me this year.”
Emerging prospect Ted Evetts clinched his first Alexandra Palace victory with a straight-sets success over Simon Stevenson.
Stevenson missed darts to win each of the first two sets and was punished by Evetts, who went on to win the third set 3-1, teeing up a second-round meeting with two-time World Champion Adrian Lewis.
Michael Barnard was Friday’s first winner, as he survived five missed match darts from Portugal’s Jose de Sousa before progressing in a sudden-death leg.
2018 PDC Unicorn Challenge Tour Order of Merit winner Barnard led by two sets before De Sousa hit back to level and then put himself in a winning position before the Englishman progressed to a meeting with Jermaine Wattimena.
Alan Tabern also overcame a scare against Australia’s Raymond Smith before securing a second-round date with world number one Michael van Gerwen following a five-set contest.
Tabern led 2-0 before seeing the 2018 DPA Order of Merit winner hit back to level, but the St Helens ace took the deciding set 3-0 to progress.
The £2.5 million event continues over two sessions on Saturday when each session again includes three first-round matches and one second round match, with the 32 seeded players joining the action in round two – including the start of tournament favourite Van Gerwen’s campaign for a third World Championship title.
Watch live on the Sky Sports Darts channel in the UK & Ireland, worldwide through the PDC’s broadcast partners or through PDCTV-HD for Rest of the World Subscribers.
2018/2019 William Hill World Darts Championship
Schedule of Play
Friday, December 14
Michael Barnard 3-2 Jose De Sousa (R1)
Alan Tabern 3-2 Raymond Smith (R1)
Kevin Burness 3-0 Paul Nicholson (R1)
Jamie Lewis 3-2 Cody Harris (R2)
Danny Noppert 3-0 Royden Lam (R1)
Ted Evetts 3-0 Simon Stevenson (R1)
Chris Dobey 3-0 Boris Koltsov (R1)
Gary Anderson 3-1 Kevin Burness (R2)
Saturday, December 15
Afternoon Session (1230)
Richard North v Robert Marijanovic (R1)
Mickey Mansell v Jim Long (R1)
Josh Payne v Jeff Smith (R1)
Max Hopp v Danny Noppert (R2)
Evening Session (1900)
Toni Alcinas v Craig Ross (R1)
Ryan Searle v Stephen Burton (R1)
Keegan Brown v Karel Sedlacek (R1)
Michael van Gerwen v Alan Tabern (R2)