The Grantown-on-Spey woman eagled the first hole in a three-way play-off to land the £10,000 top prize at The Grove in Hertfordshire.
McCook, who plays mainly on the LET Access Series, had earlier carded a four-under-par 68 to end up in a tie with English duo Georgina Blackman and Amy Taylor.
In her effort, McCook birdied the second, fourth and eighth holes in an event that had both Laura Davies and Trish Johnson in the field.
She bogeyed the ninth and tenth before making birdies at the short 13th and 16th holes and then signed off with a birdie-4 at the 18th in a strong finish.
“My golf has been feeling good, but sporadically,” said McCook, who is attached to Gleneagles Hotel. “Or signs of great filled with some inconsistency. Today it all came together, mentally also. I got out my own way, which I find so hard to do.
“I was speechless, I was so comfortable and relaxed in the play-off, I holed the putt and felt so calm and just delighted.
“My best friend, Tara Mactaggart, ran over to me and hugged me, and I cried. Just an overwhelming feeling of, I guess, dreams coming true. It’s been a rough road since I turned pro at times, so to feel all that hard work paying off, it’s just so so good.”
McCook, who is managed by Paul Lawrie through his Five Star Sports Agency, added: “I can’t thank those around me enough who have helped get me to this spot, my friends and family are so supportive.
“My sponsors have allowed me to keep going and work to get to this moment. I’ll enjoy this moment, as winning in our sport doesn’t happen often, but I’ll get back to working to feel this feeling again.”
Dryburgh won twice in the circuit’s first season in 2020 after it was set up by Justin Rose and his wife Kate to provide playing opportunities for British-based players during the Covid pandemic.
Now in its fourth year, Beveridge won the second event this season as she also prevailed in a play-off at Southport & Ainsdale.
Elsewhere, local man and defending champion Scott Mann is off to a strong start in this year’s Craws Nest Tassie at Carnoustie.
Mann fired rounds of 66-68 for a four-under total in the 36-hole qualifying to finish five shots ahead of Bristol’s Scott Wormleighton (73-66), with Primrose Green’s David Primrose (73-69) one further back in third place.
The cut for the top 64 fell at 19-over, with ten players progressing on that mark but two players missing out.