The British No 2’s 6-3 6-2 7-5 victory, which took less than two hours, booked his place in the second round for the third straight year.
Not that the 22-year-old Yorkshireman is getting carried away with his achievements.
He said: “I’m happy , but it’s only first round. It’s not like I have made semis or finals for three years in a row. So it’s good that each year I’ve definitely got better.
“My first five-set win, first grand slam match came here. I remember it was very exciting. Last year was a great match against (Nikoloz ) Basilashvili in four sets. This year I’ve won in three sets. So it’s good progression. I just felt I was the better player (yesterday) and deserved the win.”
Edmund, from Beverley, made his main draw debut two years ago as a qualifier and survived an epic evening encounter against Frenchman Stephane Robert, with Andy Murray among the spectators.
Twelve months ago he defeated Basilashvili in four tight sets before losing to John Isner.
He was left waiting to find out who his second-round opponent will be this time, with Jo-Wilfried Tsonga two sets to one down to Argentina’s Renzo Olivo.
Edmund is unusual among his countrymen in favouring clay and Portugal’s Elias was soon feeling the power of the Yorkshireman’s mighty forehand.
After racing into a 5-1 lead things got a little tricky and Elias had a chance to get back on serve, but Edmund withstood the pressure to take the first set.
He dominated the second and then fought back from 3-0 down in the third, clinching victory with his 24th winner to join Murray and Aljaz Bedene in making it three British first-round winners.
Johanna Konta insisted she is happy with the progress she has made through the clay season despite crashing out of the French Open in the first round.
The seventh seed appeared to have a kind draw against world No 109 Hsieh Su-wei as she sought her first main-draw win at Roland Garros.
Konta raced through the first set, but then got bogged down in a battle and became thoroughly frustrated as Hsieh ground out a 1-6 7-6 (7/2) 6-4 victory.
Konta is not nearly as at home on clay as on hard or grass, but this was still a disappointing result.
The 26-year-old said: “My expectations are never results-orientated. Whether I’ve lost first round or third round or quarters, I do my best to not judge myself on that aspect.
“Obviously, I like to be involved in a tournament for as long as possible, and I would have liked to have been involved here longer. But, in terms of the match that I played and the improvements that I feel I’ve made that I have spoken to my team about, we all seem to be on the same page.
“I do feel I’m a better player at the end of these last few weeks on the surface than I was at the beginning and that is the most I can ask of myself: to keep improving.
“If that doesn’t always come through in results, that’s part of sport. I played a player who played incredibly well, who was inspired, and that’s also something that I’m also very happy to be a part of because it gave me another opportunity to learn about myself and see the things that I can do better.”