World number 47 Gojowczyk failed to convert any of his five break points on the fast Ohio hardcourts, easing the pressure on the 20-times grand slam champion.
Federer, playing in his first match since turning 37 earlier this month, said taking the fight to his opponent was a key to his continued success.
"It's important to always try to play your game, especially on faster surfaces, and not be too reactive," he said.
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"In this day and age everybody serves big, everybody can hit big from the baseline... so it's important to bring in your own energy, whether it's the first round or the finals."
Federer will look to continue toward a record eighth title at the Ohio event when he faces Argentine Leonardo Mayer in the third round on Thursday.
TSITSIPAS FLAMES OUT
Twenty-year-old Tsitsipas was unable to build on his magical Toronto run as the Greek upstart fell 7-5 6-3 to Belgian David Goffin.
Tsitsipas, who strung together four wins over top-10 players in Toronto before falling to Nadal in Sunday's final, converted just one of eight break points.
Goffin avenged a quarter-final loss to world number 15 Tsitsipas in Washington two weeks ago and will next face Frenchman Benoit Paire.
South Korean Chung Hyeon rallied to beat Jack Sock 2-6 6-1 6-2 and set up a second-round clash with fourth seed Juan Martin del Potro.
Australian Open semi-finalist Chung, who in April became the first South Korean to crack the top 20, won 70 percent of his first-serve points en route to the win.
"It was a really tough match," he said. "In the first set he had a really good serve and big forehand but I was just trying to stay strong all the time and stay calm as well."
The clash with Del Potro, who this week rose to a career-high third in the world rankings, will be their first meeting.
Sock's defeat was his eighth in succession dating back to the Italian Open.
Third seed Alexander Zverev will meet Robin Haas in the second round, while Milos Raonic advanced to face Tunisian lucky loser Malek Jaziri.
(Reporting by Rory Carroll in Los Angeles and Frank Pingue in Toronto, editing by Pritha Sarkar / Ian Ransom)
