The Swiss produced a majestic display to show why he is favourite to win his sixth Australian Open and 20th Grand Slam title at the age of 36.
Bedene, who has switched his allegiance back to Slovenia, played well but the result was never in doubt.
Federer won in 99 minutes and will face Germany's Jan-Lennard Struff next.
He broke Bedene's serve in the fourth game of the first set in which he conceded only three points on his serve.
The five-time Australian champion pounced immediately at the start of the second to gain another break and eased through the rest of the match, finishing with 41 winners against a player who has now made six appearances in the tournament without a win.
Federer is seeded second behind Rafael Nadal, whom he beat in last year's final, but with doubts over the Spaniard's fitness plus other long-term rivals Novak Djokovic and Stan Wawrinka, the Swiss is seen as the man to beat this year.
"I am hoping for another good year," said Federer, who also won Wimbledon for a record eighth time in 2017.
"I am not sure it will go that well because last year went so well and I'm a year older, Rafa's looking in tip-top shape and others guys are coming back.
"Last year was a fairytale and I can't control everything. Last year could be my favourite year of my career but I am hoping to stay healthy, give myself chances and play my very best in the big matches like last year."
Will Ferrell interviewed Roger Federer after the Swiss' win
Will Ferrell joined John McEnroe to interview Federer in the guise of his alter ego Ron Burgundy
The only time Federer looked remotely stumped came during a post-match interview conducted by American comedian Will Ferrell who told him he had played like a "silky gazelle".
"Don't they get eaten in the end?" replied Federer before being asked whether he was a witch or a vampire and whether his age-defying performances were down to a diet of wombat meat.
