Federer had skipped the clay court season the past two years since last competing in Rome in May 2016 but confirmed his participation for the tournament in Madrid earlier this year.
Carlos Alcaraz came through a testing third round encounter against Nicolas Jarry.
Britain battled to victory over last year's Davis Cup runners-up Australia in their Finals group stage opener on Wednesday, while Spain and Italy both suffered chastening defeats at home in the premier men's team competition.
'Rarely did I feel my best on clay in the first or second tournament in the season. That was the case this season again. It took me two tournaments to really feel that I'm getting closer to the desired level.'
Defending champion Stefanos Tsitsipas came out on top from a thrilling quarter-final clash with Diego Schwartzman, prevailing 6-2, 6-7(3), 6-4 on Friday to set up a meeting with Alexander Zverev at the Monte Carlo Masters.
Novak Djokovic enjoyed a stress-free return to US competition following a two year absence
'There is so much excitement brewing about Wednesday's Alcaraz and Rune quarter-finals showdown -- it could well be a juicy foretaste of things to come.'
A summary of Sunday's action at the US Open.
Carlos Alcaraz crashed to a humbling 6-3 7-6(4) defeat by Hungarian qualifier Fabian Marozsan in the third round of the Italian Open.
Images from Day 8 of the 2021 French Open at Roland Garros in Paris on Sunday
Serena will not be in the US team for the Tokyo Olympics
Images from Day 7 of the 2020 US Open in New York on Sunday.
IMAGES from Day 1 of the Wimbledon Championships played on Monday.
Not since Stan Wawrinka's 2016 US Open triumph has a Grand Slam been won by someone other than the 'Big Three', winners of the last 13 major titles.
India ended its campaign at the Fed Cup with a fourth place finish after losing the classification tie 1-2 to Korea as the team badly missed injured Karman Kaur Thandi, who did not play on Saturday, in Astana, Kazakhstan.
Earlier, title holder Borna Coric beat American Sebastian Korda 7-6 (5), 6-4, while another home hope Taylor Fritz survived a first-set thriller to beat Czech Jiri Lehecka 7-6 (14), 6-2.
Holder Stefanos Tsitsipas moved into the Monte Carlo Masters quarter-finals with a 7-5, 7-6(1) victory over Laslo Djere on Thursday while fourth seed Casper Ruud was knocked out in straight sets by Grigor Dimitrov.
Frances Tiafoe, Taylor Fritz, Tommy Paul and Coco Gauff all advanced to the third round of the US Open on Wednesday to keep home hopes alive.
A round-up of all the tennis action from around the globe.
Tennis matches at the Ariake Tennis Park will begin from 3pm local time (0600 GMT) from Thursday, governing body International Tennis Federation said after requests from players who had struggled in the stifling weather.
Images from Day 3 of the 2024 French Open at Roland Garros in Paris on Tuesday.
Holder Stefanos Tsitsipas moved into the Monte Carlo Masters quarter-finals with a 7-5 7-6(1) victory over Laslo Djere
IMAGES from Day 5 of the US Open, at the USTA Billie Jean King National Tennis Center, Flushing Meadows, New York, on Friday.
Coco Gauff rallied from a break down in the final set to beat Swede Rebecca Peterson and reach the Indian Wells quarter-finals for the first time
Images from Day 10 of the 2021 French Open at Roland Garros in Paris on Tuesday.
WTA chief promises 'strong reactions' to ban on Russian, Belarusian players
Images from Day 4 of the 2023 Australian Open, at Melbourne Park.
Summary of sports events and persons who made news on Tuesday
A summary of Friday's action at the US Open.
Images from Day 3 of the 2020 French Open at Roland Garros, in Paris, on Tuesday.
Summary of sports events and persons who made news on Saturday
IMAGES from Day 5 of the 2020 US Open in New York on Friday.
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