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Seventh seed Riske, ranked 35, is a former Wimbledon quarter-finalist and was runner-up at the WTA Tour event in Nottingham last week but she had no answer to the clean power of Boulter.
The British number four, who is ranked more than 100 places lower at 141, is known as a big-match performer and she showed that again at Edgbaston, winning 6-4 6-3.
Boulter, who found out just before going on court that she has been given a Wimbledon wild card, said: “Massive milestone, definitely excited of course.
“I feel like I’ve got close to a few of the top players and never got over the line. I feel like I can do it, the game is there, and that’s what’s exciting to me.
“I feel I can go out on the court and compete with anyone out there. A few of those under my belt and really kick on and I feel like I’ll be in a good place.”
Boulter has so far been prevented from achieving her potential by injuries and illness, and a leg problem suffered in March sidelined her for nearly three months.
The Leicestershire player is determined to enjoy the chances she has, particularly at home on grass, and said: “These past few years have taught me a lot.
“I just want to go out there, smile, have fun, maybe not take it too seriously, and I think that really works for my game.
“I felt like I did that out there and I felt very comfortable, and the work that I did out there started to show. I believe it’s going to come out. Hopefully I can build on that from here.”
The 25-year-old made the perfect start with a break of the Riske serve in the opening game and survived her only moments of real pressure by saving three break points in the fourth game of the second set.
She finished the match off in style with a run of four games in a row to set up a second-round clash with France’s Caroline Garcia.
Top seed Jelena Ostapenko recovered from a dramatic collapse against Canada’s Rebecca Marino to win 6-2 5-7 6-3 having led 6-2 5-1.
The Latvian earned a few boos from the Edgbaston crowd for her post-match comments, saying: “The biggest opponent to me was myself because I played really well until 6-2 5-1, she didn’t really do anything, she didn’t change anything, then I started to fight against myself, I started to miss some balls.
“I’m glad I won because my level is way better than her today and I showed it.”
Fourth seed Elise Mertens fell 3-6 6-4 7-5 to American Caty McNally while sixth seed Sorana Cirstea beat Shelby Rogers 6-3 4-6 6-1 and Ukraine’s Dayana Yastremska was a 6-4 6-3 winner over Magda Linette.