The Swiss third seed wasted little time in disposing of the Georgian 6-2, 6-1, 6-2 in a night match on Rod Laver Arena to set up a second round meeting with practice partner Aleksandr Dolgopolov of Ukraine.
The 17-time Grand Slam champion was in a different class to his 117th-ranked opponent and showcased his full arsenal of shots against the bemused Basilashvili in their first meeting.
"That was a good match. I'm really pleased how I was able to play," Federer said.
"It definitely gives me a bit of a lift in confidence because this year I haven't been able to play properly yet," he said.
"I mean, I had some decent matches in Brisbane, but it was all under a sort of a cloud knowing that I wasn't 100 percent.
"But this was a match where I was able to focus on my game, on tactics, all that stuff. So it was nice to play that way."
An illness hampered Federer at the lead-up Brisbane International tournament where he lost in the final to Canada's Milos Raonic.
Federer broke the Georgian eight times in a dominating performance and maintained his record of never having lost in the first round in Melbourne in 17 appearances.
The Basel native, playing in his 65th consecutive Grand Slam tournament, hit a total of 31 winners and won 91 percent of his first serve points.
Federer's next opponent will be the quirky shotmaker Dolgopolov, who he practised with during the off-season.
"I think it's going to be very tough," Federer said.
He noted he had practised with Dolgopolov in the off-season in Dubai.
"I know him very well," he said.
"This is going to be a different challenge than the first round.
"This was more of an unexperienced player today, but Dolgopolov is a different player, a different level.
"He's been there before. He's got the fitness, the power, the speed, tennis IQ, all that. It's going to be a big challenge."
Federer is looking to claim his fifth Australian Open crown and at 34 become the oldest man to win the Grand Slam since Ken Rosewall at 37 when he lifted the trophy in 1972.
The Swiss legend is on course for a possible meeting with top seed and five-time winner Novak Djokovic in the semifinals as he looks for his first Grand Slam title since Wimbledon in 2012.