Italy’s Sinner struggled with a hip injury in the build-up to Roland Garros but he moved well in a routine 6-3 6-3 6-4 win.
Sinner only looked troubled when serving out the match, handing his American opponent two break points with a double fault before recovering to seal victory.
“The hip is good, I am very happy,” the 22-year-old said.
“The general shape is not at 100% yet so we try to build every day.”
Greek ninth seed Stefanos Tsitsipas also won on Monday, beating Marton Fucsovics 7-6 (9-7) 6-4 6-1, despite much of the day’s play being held up by rain.
And there were also victories for Russian fifth seed Daniil Medvedev and Frenchman Gael Monfils in the evening session on day two.
Sinner will become world number one in Paris should defending champion Novak Djokovic fail to reach the semi-finals.
He put in a strong showing against Eubanks, hitting 33 winners and saving nine of the 10 break points he faced.
He will face home favourite Richard Gasquet next.
“I’m happy about what I’ve achieved in the last months. The goal is to improve every day,” he added.
Tsitsipas is still seeking a first major title, having let a two-set lead against Novak Djokovic slip in the 2021 final.
After a slow start to the season, he won his third Monte Carlo Masters title and reached a second clay-court final in Barcelona, losing to Casper Ruud.
He was made to work against veteran Fucsovics, saving three break points and a set point on his way to clinching the opener.
After another tight second set, Tsitsipas ran away with the third, wrapping it up in 31 minutes to set up a second-round meeting with Germany’s Daniel Altmaier.
“I felt lucky to walk away with the first set and I brought that confidence into the second one,” the 25-year-old said.
“I played brilliant tennis towards the end. I’m really happy to insert myself in that sort of way.”
Elsewhere on day two, Australian Open finalist Medvedev reached the second round for just the third time by defeating German Dominik Koepfer 6-3 6-4 5-7 6-3.
Frenchman Monfils beat Thiago Seyboth Wild of Brazil 6-2 3-6 6-3 6-4 on Court Philippe Chatrier.
Tommy Paul, seeded 14th, cruised to a 6-2 6-3 6-1 victory over Argentine Pedro Cachin, and American 15th seed Ben Shelton came from a set down to get past Frenchman Hugo Gaston 3-6 6-3 6-4 6-4.
