Twenty-time Grand Slam winner Federer took just 52 minutes to see off Frenchman Gasquet 6-2 6-3 in what was his first match on clay in three years after skipping the 2017 and 2018 seasons on the surface to manage a knee injury.
"I would have always felt regrets not being on the clay in 2019 because my knee problem is far enough away now. I grew up on this surface, so I feel comfortable on it," the 37-year-old Federer told reporters.
"It feels good to be back on the clay now and I enjoy it. Some of these rallies where you get pushed to the side, you slide, you hit the ball, you slide, you hit the ball, you come back into it, then you can defend in a different manner than you do on the hard courts, or on the grass."
World number one and Australian Open champion Djokovic needed only one hour and five minutes to win 6-4 6-2 against 21-year-old qualifier Fritz, ranked 57th in the world, who he also beat in straight sets at the Monte Carlo Masters last month.
Last year's runner-up Dominic Thiem reached the round of 16 without having to finish his match with American Reilly Opelka, who withdrew due to injury one game into the decider.
Austrian Thiem lost the first set 7-6(2) before winning the second 6-3 and was leading 1-0 when 21-year-old Opelka retired.
In first round matches, Monte Carlo Masters champion Fabio Fognini comfortably beat Britain's Kyle Edmund 6-4 6-3 to reach the last 32, while Stan Wawrinka had a straight sets win over Pierre-Hugues Herbert.
Home favourite David Ferrer, who is playing the last tournament of his career, delayed his retirement by beating fellow Spaniard Roberto Bautista Agut 6-4 4-6 6-4 and will play German hotshot Alexander Zverev in the round of 32.
Five-time Madrid champion Rafael Nadal enters the tournament on Wednesday when he faces Canada's Felix Auger Aliassime in the second round bidding to reach the last 16.
(Reporting by Richard Martin; Editing by Ken Ferris)
