The Russian advanced with an impressive victory over home favourite John Isner 6-4, 7-6.
Fourth-seeded Federer, fresh off his 100th career title, looked poised to finish off Edmund in less than an hour. The 32-year-old Spain-born has won 17 Grand Slam titles as well as bagged 33 ATP Tour Masters 1000 titles.
The Austrian seventh seed managed to break once in each set, while losing just seven points on serve for the match.
"He's been around for so many years and he doesn't really get overwhelmed by the occasion of playing on center court", he told reporters.
The 24-year-old, now ranked 23 in the world, won 84% of his first-serve points to triumph 6-3 6-3 in an hour and 14 minutes.
The 37-year-old Swiss will be up against a newcomer to the tournament quarterfinals, Poland's Hubert Hurkacz, who ousted 24th seed Denis Shapovalov of Canada 7-6 (3), 2-6, 6-3.
Speaking of their first meeting, Federer admitted the conditions of the match were tough but said Edmund still had what it takes to achieve big results.
"I was relieved I got through that one", Federer said.
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Kecmanovic, ranked 130th in the world, had taken the first set against Yoshihito Nishioka 6-4 when the 74th-ranked Japanese retired because of injury.
Svitolina beat the Czech Republic's Marketa Vondrousova 4-6, 6-4, 6-4 in the only other women's singles match of the day.
Kecmanovic is the last teen left in the men's draw, after the departure of 19-year-old Shapovalov.
"The previous days here in Indian Wells helped me a lot to get back to my top level physically and also to play good tennis again", Thiem said.
"I felt pretty confident on my own service games".
Edmund did have moments of joy on the Federer serve - most notably forging three break chances at 5-4 down - but at match point a searing backhand down the line from the world number four meant the Brit could only net and relinquish any hope of an unlikely comeback.