Elliott joined the Gladiators in September following the departure of Paul Phillips.
Buxton chairman Dave Hopkins asked Matlock chairman Jay Beaumont for permission to speak to Elliott on Friday, and on Sunday afternoon the former Boston United boss told the club that he would be accepting Buxton’s offer.
Elliott will replace Jamie Vermiglio, who resigned due to increasing personal commitments.
He joins a side currently 18th in the National League North following their promotion from the Northern Premier League as title-winners last season.
John McDermott and Richard Boryszczuk, Elliott's assistants at Matlock, have also left the club to join him at Buxton.
On Elliott's departure, chairman Jay Beaumont said: “Whilst having only known Craig for three months, it has been very obvious what a good manager and person he is, and it was no surprise when I received a call asking for permission to talk to him.
“I’m a believer that in life you can’t stand in the way of people wanting to progress.
“I did tell him that undoubtedly I wanted him to stay on as manager of our great club but at the same time I was happy for him to talk to Buxton. The decision had to be his.
“He’s worked tremendously hard during the past three months and I’m convinced he was rebuilding a very good side for us.
“I’d like to thank him for all that he’s done and wish him well for the future.”
The Gladiators will begin the search for a successor to Elliott ahead of next Saturday’s trip to Stalybridge Celtic.
In the meantime, the club have announced that goalkeeping coach Steven Shuttleworth and captain Ryan Qualter will take charge of training.
Photo credit: Buxton
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