The drizzle returned at 4-4 in the second set but Federer switched gears after the restart, sealing a decisive break to go ahead 5-4 and closed out the match in an hour and 56 minutes.
The 20-times Grand Slam winner, who is seeking his 100th ATP title, next takes on the winner of the quarter-final between sixth seed Borna Coric and Georgia's Nikoloz Basilashvili.
Gael Monfils continued his fine recent form by rallying past Lithuanian qualifier Ricardas Berankis 6-1 6-7(3) 6-2, with the victory set to catapult him back into the top 20 on Monday.
The Frenchman, who won his first title of the year in Rotterdam earlier this month, takes on Australian Open semi-finalist Stefanos Tsitsipas in the last-four.
Tsitsipas, 20, survived a mid-match wobble to get past Polish qualifier Hubert Hurkacz 7-6(4) 6-7(1) 6-1.
"We're both serving really well. We have similar game style. I'm a bit more aggressive than him, but he's much faster," Tsitsipas said of Monfils.
The pair have a 1-1 head-to-head record with world number 23 Monfils winning their most recent meeting in Sofia this year.
"I'm going to have to... be patient, play with passion as well, wait for the opportunities to break him. I'm going to have to serve well," Tsitsipas added.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Toby Davis)
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