Raducanu entered the match with a 0-3 record against Kasatkina and she had not previously won a set against the 28-year-old Russian-born Australian star.
The 22-year-old broke Kasatkina four times as she stormed to win a one-sided first set in under half an hour.
After Raducanu held in the opening game of the second set, there were six consecutive breaks of serve before the Brit converted her fourth match point to serve out for a convincing win.
While Raducanu lost serve four times, she broke in all eight of her opponent's service games.
Raducanu, who started the week as the world No 43, is up two places to 41st with the 60 points earned for her first round win in Strasbourg.
The former world No 10 could return to the top 40 if she reaches the quarter-finals of the 500 tournament.
Raducanu said after her victory: "I'm really proud of my performance. How I came out and I was really aggressive.
"I also managed to maintain my focus in the second set when it was getting close and Daria was playing some great points when I was trying to serve the match out so I'm really proud of today's performance.
"I think I've been building towards this. It's a great feeling because I think I had three great matches in Rome and the fourth I played an opponent who was better than me on the day.
"To kind of bring that level here, I'm really proud. But I think as I spend more time on clay, I'm starting to like it more and more. I'm building my relationship with the surface."