The third seed Swiss beat his 189th ranked opponent 7-6 (7-5) 7-6 (7-3) 6-3 to keep his campaign alive for winning a record extending 21st Grand Slam. However, Evans earned quite a few admirers in the Rod Laver arena courtesy his his depth and precision with his shot-making.
Federer is scheduled to play Taylor Fritz in the third round after the American inflicted a major upset beating France's Gael Monfils.
"I couldn't pull away early in the match - it helps when you can sneak in a quick break," said third seed Federer, who at 37 is aiming to become the first man to win seven Australian Open titles. "He played very well. It was hard to pull away, to his credit. I thought I played well."
British number four Evans, ranked inside the top 50 before a drugs ban, is aiming to make a leap in the singles rankings as he continues his comeback. And the 28-year-old showed he still has all the tools to cause problems against the world's best.
Federer was particularly impressed with Evans who, like the Swiss great, has a game heavily reliant on a sliced backhand. In their previous two meetings, Evans had never won more than five games in a set off Federer , bettering that on Rod Laver as he took Federer into a first-set tie-break.