The second-round success in 80 minutes came just over a week after Federer beat the Uruguayan on clay in the final in Istanbul.
Federer was tested in Rome all the way by Cuevas, ranked 24th in the world, with the second seeded 33-year-old winning a tiebreak to seal the opening set in 45 minutes.
A break for 3-2 in the second gave Federer — seeking his first title in Rome after losing three times in the final in 14 previous appearances — the
lead which he never surrendered.
Scotland's Andy Murray kept his clay-court purple patch alive, defeating Jeremy Chardy 6-4, 6-3 to extend his unbeaten record on the surface this season.
Murray extended his record to 10 wins in 10 matches after making a late decision to see how his body had held up after the stunning Madrid Open win on Sunday over Spaniard Rafael Nadal.
Murray, who turns 28 on Friday, beat Chardy in 84 minutes with six aces and three breaks of the Frenchman's serve.
“My legs were a bit tired at the start but they started to feel better after a few games,” said Murray.
“He (Chardy) has a big serve, it was tough for me.” he said.
“But I hung in, moved well and didn't give him loads of opportunities.”
Nadal, his seeding down to fourth, began his title defence and the quest for a record eighth Rome trophy as he crushed Turk Marsel Ilhan 6-2, 6-0 in barely one hour and 15 minutes.
“I'm playing better,” said Nadal.
“It's true that I've not played two weeks in a row at the right level yet — but today was a good start.”
Japanese fifth seed Kei Nishikori won his opening test four days after losing to Nadal in the Madrid semi-finals, with Asia's top player defeating
Czech Jiri Vesely 7-6 (7/3), 7-5
Italy's Fabio Fognini thrilled the home fans as he upset tenth seed Grigor Dimitrov 7-6 (11/9), 4-6, 6-0, reaching the third round here for the first
Dominic Thiem beat 12th seed Gilles Simon as the French player quit with neck pain when trailing 7-6 (7/5), 2-0.