Britain's Raducanu recovered from being broken twice in the first set before demolishing her opponent in the second set to set up a second-round match against world number five Jessica Pegula.
Raducanu, who won the US Open as a qualifier in 2021, has slipped to 168th in the world after an injury-plagued 2023 season.
The 21-year-old missed the French Open but returned to competition at the Nottingham Open grass-court tournament earlier this month, where she reached the semi-finals.
"I think for me it was the case of adapting to the court," said Raducanu, who has been given a wild card for Wimbledon, which starts on July 1.
"I've never played in Eastbourne, I've never played on this centre court and every court plays a little bit differently. In the beginning, I was trying to figure out what was the best way for me to play and use my game."
Stephens, who won the US Open in 2017, broke first for a 3-2 lead but failed to hold serve for the rest of the match.
Raducanu, after saving two break points to hold for 5-4, broke the American for a third time to seal the first set and then sprinted to the finish line, completing victory in 76 minutes.
Also in first-round action on England's south coast, Elise Mertens beat Yuriko Miyazaki 6-1, 6-2 while Ashlyn Krueger overcame Viktorija Golubic 6-1, 6-7 (7/9), 7-5.
In the men's tournament, Britain's Cameron Norrie continued his woeful run of form, losing his first-round match against Finland's Emil Ruusuvuori 7-6 (11/9), 6-3.
Brazilian Thiago Seyboth Wild also advanced with a 6-4, 6-4 win against Australia's James McCabe to set up a second-round showdown with top seed Taylor Fritz.
