Despite the straight-sets win, Evans challenged the 2018 champion throughout, leading 5-3 in the first set tiebreak at one point and bouncing back from 5-3 down in the second set to force another tiebreak.
"I couldn't pull away early in the match. It always helps when you sneak in a quick break, win that first set," Federer, who is bidding for his 21st grand slam title, said in an on-court interview.
"I think maybe if I'd been able to close out that second set things are maybe going a bit smoother but I think he played very well."
Meanwhile, Tiafoe, who was competing at his first Australian Open, pulled off a shock victory over South Africa's Anderson 4-6, 6-4, 6-4, 7-5.
In his excitement post-match, the world number 39 got his necklace caught up in his shirt and had to be freed by a ballkid.
"It means the world to me," Tiafoe said of the win. "I lost to Kevin three times last year and I was down a set and a break and it looked like he was going to do it for the fourth time."
Asked about his 21st birthday, which he celebrates on Sunday, he said: "If I'm playing on that day and beat somebody that would be the best present I could get."
Croatian sixth seed Marin Cilic also seized a 7-5, 6-7 (9-11), 6-4, 6-4 win over Mackenzie McDonald of the United States, and rising star Stefanos Tsitsipas of Greece posted a 6-3, 2-6, 6-2, 7-5 victory over Serbian qualifier Viktor Troicki.
In the women's draw, Sloane Stephens demolished Hungary's Timea Babos 6-3, 6-1 and local favourite Ash Barty took a comfortable 6-2, 6-3 over China's Yafan Wang.
Stephens, the 2017 US Open champion, took just an hour and 35 minutes to dismantle Babos.
"We actually won three junior doubles grand slams together so I know that she is a very big player and I just had to be patient," Stephens said post-match.
"Her serve is ridiculously good, so I just had to hang in there."
But there was an upset for ninth seed Kiki Bertens of the Netherlands, who was stunned 3-6, 6-3, 6-3 by Russian world number 44 Anastasia Pavlyuchenkova.
Twentieth seed Anett Kontaveit was also dumped out of the tournament 6-3, 6-3 by Belarusian Aliaksandra Sasnovich.
"I couldn't pull away early in the match. It always helps when you sneak in a quick break, win that first set," Federer, who is bidding for his 21st grand slam title, said in an on-court interview.
"I think maybe if I'd been able to close out that second set things are maybe going a bit smoother but I think he played very well."
Meanwhile, Tiafoe, who was competing at his first Australian Open, pulled off a shock victory over South Africa's Anderson 4-6, 6-4, 6-4, 7-5.
In his excitement post-match, the world number 39 got his necklace caught up in his shirt and had to be freed by a ballkid.
"It means the world to me," Tiafoe said of the win. "I lost to Kevin three times last year and I was down a set and a break and it looked like he was going to do it for the fourth time."
Asked about his 21st birthday, which he celebrates on Sunday, he said: "If I'm playing on that day and beat somebody that would be the best present I could get."
Croatian sixth seed Marin Cilic also seized a 7-5, 6-7 (9-11), 6-4, 6-4 win over Mackenzie McDonald of the United States, and rising star Stefanos Tsitsipas of Greece posted a 6-3, 2-6, 6-2, 7-5 victory over Serbian qualifier Viktor Troicki.
In the women's draw, Sloane Stephens demolished Hungary's Timea Babos 6-3, 6-1 and local favourite Ash Barty took a comfortable 6-2, 6-3 over China's Yafan Wang.
Stephens, the 2017 US Open champion, took just an hour and 35 minutes to dismantle Babos.
"We actually won three junior doubles grand slams together so I know that she is a very big player and I just had to be patient," Stephens said post-match.
"Her serve is ridiculously good, so I just had to hang in there."
But there was an upset for ninth seed Kiki Bertens of the Netherlands, who was stunned 3-6, 6-3, 6-3 by Russian world number 44 Anastasia Pavlyuchenkova.
Twentieth seed Anett Kontaveit was also dumped out of the tournament 6-3, 6-3 by Belarusian Aliaksandra Sasnovich.