Barcelona and Madrid champion Alcaraz, who is chasing his fifth title of 2023 and fourth on clay ahead of Roland Garros, overcame an early loss of serve amid heavy conditions at the Foro Italico to level up the opening set at 2-2.
Alcaraz, who had a first-round bye, earned another chance to break Ramos-Vinolas with a deft drop shot in an intense 10th game and took the advantage in the match when his compatriot finally cracked under the pressure and sent a forehand long.
Alcaraz continued to build momentum and raced ahead 3-0 in the second set against a fading Ramos-Vinolas, wrapping up the match on serve to make it 12 wins in a row.
"It wasn't easy," said Alcaraz, who will now leapfrog Novak Djokovic into top spot in the rankings on May 22nd.
"The first round of every tournament is really tough and, of course, Ramos-Vinolas is a specialist on clay. I had to adapt my game quickly to be able to go through.
"The conditions weren't easy, the rain and the waiting all day to know if I'm going to be able to play... but I'm really happy with my performance in the end."
Monte Carlo champion Andrey Rublev earlier overcame Alex Molcan 6-3 6-4 in a twice-interrupted match to book his place in the third round.
"I'm happy that I was able to finish just before the rain," Rublev said as the showers arrived again following the clash. "I can rest now and I can rest tomorrow. I don't need to stress waiting until the afternoon."
Rublev's Russian compatriot Daniil Medvedev will not have that luxury, however, and will look for his first victory in Rome when he opens his campaign against Finn Emil Ruusuvuori on Sunday after rain forced play to be suspended late in the evening.
Alexander Zverev, the 2017 champion, will also be in action against David Goffin, while fifth seed Stefanos Tsitsipas will resume his match with Nuno Borges from 6-3 4-3.
Qualifier Camila Osorio served up another shock in the women's draw after her win over 29th seed Petra Martic, when the Colombian downed world number four Caroline Garcia 6-4 6-4 to make the round of 16.
Sofia Kenin's resurgent run came to an end as the former Australian Open champion, who took out world number two Aryna Sabalenka in the previous round, failed to find a way past Anhelina Kalinina to lose 6-4 6-2.
China's Wang Xiyu reached her first round of 16 at a WTA 1000 event by edging past Taylor Townsend 6-2 0-6 7-5.
