Sinner, the Australian Open champion, withdrew from the Madrid Open and missed the Rome Masters with a hip injury, but was back on form at Roland Garros as Eubanks exited at the first hurdle for the second successive year.
"The hip is good, I'm glad that my team and myself, we were working very hard to be on court as soon as possible," Sinner said.
"For sure, the general shape is not at 100% yet, so we try to build every day."
The Italian broke twice in the opening set to lead 4-1 before Eubanks broke back but Sinner saw out the set with another break, and needed just one break in each of the next two sets to take his place in the second round.
"Honestly, I'm just happy to be back on court, I was injured, so I'm very happy to be back here," Sinner said.
"It's a very special tournament for me, it was the first time in a quarter-final of a grand slam, so I have some great memories."
The American fought bravely to the end, forcing two break points with Sinner serving for the match, but the Italian's smile soon returned as he recovered under the roof on Court Suzanne Lenglen.
"Thanks everyone, the support has been amazing, the roof closed for the first time here on Suzanne, so I'm very happy," Sinner said.
Sinner, who can become world number one for the first time over the next fortnight, will next play French wildcard Richard Gasquet, the Italian hoping to avoid a repeat of last year when he went out in the second round.
"I know I have to improve some things so I'm happy that I have my team. They push me every day and let's see what I can achieve in the future," Sinner said.
"Also, being happy on the court, which is maybe the most important, and I truly enjoy to play tennis so I'm very happy to be here."