Having beaten Rafael Nadal and Novak Djokovic earlier this month, Alcaraz has been flagged as a potential underdog to win at Roland Garros.
But the sixth seed came close to a second-round early against Ramos-Viñolas – first when he faced match point in the fourth set, and then again when he lost 0-3 in the fifth.
“I believed in myself all the time,” Alcaraz told reporters. “Of course it was hard… but I believed in myself.”
It took courage and determination for Alcaraz to stay in the match and on numerous occasions he raced across the court to save points that seemed to have been lost.
He can look back and think of the 23 break points he missed, while Ramos-Vinolas was far more focused, winning six of his seven break points.
But Alcaraz, who next faces American Sebastian Korda in the third round on Friday, scored 74 winners – thanks in large part to his excellent forehand – in the four hours and 34 minutes of the meeting.
“I’m still young, but I would say that I have already become an experienced player,” said Alcaraz.
“I feel comfortable playing in big stadiums, big games, grand slams. Physically I am strong, I am also strong mentally, so I think I am ready to play such matches in such situations, in these tournaments.
Another top five-setter in Wednesday’s men’s draw was third seed Alexander Zverev, who also saved match points by beating Sebastian Baez 2-6 4-6 6-1 6-2 7-5.
Zverev broke four times in the first two sets before recovering with four breaks of his own in the third and fourth.
A tense fifth set seemed to go in favor of Baez as he made a break and then scored a match point at 5-4, but Zverev, last year’s French Open semi-finalist, was able to fight back and secure the decisive break. 5-5.
“We just need to find a way,” Zverev said. “You talk about fortitude, you talk about some players – the greats, Roger (Federer), Raf (Nadal), Novak (Djokovic) – they always find a way out in the most difficult moments.
“That’s why they are the way they are. I’ll never be at their level, but I’m trying to get close to them and that’s definitely something you need to do.”
Nadal and Djokovic had more direct wins in the second round, with the former seeking their 14th French Open title by defeating Corentin Moutet 6-3 6-1 6-4, and the latter claiming their third title by defeating Alex Molkan. 6-2 6-3 7-6.
Nadal, who was playing in his 300th Grand Slam match, suffered the injury months after winning the Australian Open in January. “I just enjoy being at Roland Garros, the most important tournament of the year for me,” he said.
Djokovic, meanwhile, appeared to be heading for the third round against Molcan, but was forced to play a third-set tiebreaker in what he said were windy and difficult conditions.
He will next face Aljaz Bedene on Friday as he continues to defend his title.
In the women’s draw, reigning US Open champion Emma Raducanou said she views every loss as a “lesson” to see “where I went wrong, where I can improve” as she lost 3-6 6-1 6-1 against Alexandra Sasnovich . .
In her first clay court swing, the 19-year-old Radukanou struggled with injuries ahead of her French Open debut, but also said she “got stronger” on the surface as the season progressed. She then turns her attention to grass and her home Wimbledon Grand Slam next month.