Playing with little inhibition and plenty of power, Khachanov broke Federer early on before the Swiss switched gears to march ahead and take the opening set in 29 minutes.
Federer, who beat Khachanov in straight sets at Halle last year in their only previous meeting, showed his experience in the second-set tie breaker that swung back and forth and clinched the contest on his third match point.
"It was nice to see him free swinging the entire match, which isn't easy to do," Federer said of his opponent.
"I remember those days, match point down, you go for it, you don't care. That is what you have to remind yourselves sometimes when you're older, to go for it. He mixed up the play well, so it was a tough match for me.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"The tiebreak was close, it was good to play that early on in the season, to play those nervy situations with the pulse up high, you're sweating loads. The match had a bit of everything, so I'm very happy with the performance."
Bencic had a tougher outing against world number 15 Pavlyuchenkova but prevailed 6-1 3-6 6-3 before teaming up with Federer to win the mixed doubles dead rubber.
Earlier, the United States scored a 2-1 win over Japan after Naomi Osaka was too ill to participate. Osaka's withdrawal gave CoCo Vandeweghe a 6-0 6-0 victory in their singles match.
Japan drew level after Jack Sock retired while trailing against Yuichi Sugita but Osaka's absence meant the Americans were awarded a 4-0 4-0 win in the deciding mixed doubles rubber.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Christian Radnedge)
