The 39-year-old has been an influential figure at Headingley since joining the county as first-team coach in November 2011, and led them to the LV= County Championship title last year.
While his coaching position with the White Rose is unaffected, he has signed a two-year deal with the South Australian franchise in a move that had been widely anticipated.
He will begin his role there in December, when the next edition of the BBL gets under way.
Gillespie, who played for South Australia during the entirety of his playing career, said: “I would like to place on record my gratitude to Yorkshire County Cricket Club for allowing me to coach in the Big Bash League.
“I am looking forward to bringing back some knowledge and experiences to Yorkshire, which will improve our T20 cricket going forward.
“I am excited to be taking charge of my hometown team in the Adelaide Strikers. Their initial interest came a few weeks ago and after a few conversations I am pleased that we came to an agreement.
“The opportunity to coach and build on the Strikers’ recent success is a great challenge.”
Gillespie will take up his new post with the full blessing of Yorkshire, who have never won England’s domestic Twenty20 competition.
Chief executive Mark Arthur said: “I want to make it absolutely clear that Jason (Gillespie) is primarily our senior coach and he will join Adelaide Strikers with our blessing for a short period at the end of the year. It will advance his own skill-set, which ultimately will benefit Yorkshire.
“Over the last 10 to 12 years we have underperformed in the T20 competition. This year we have made it a priority. We will be making a concerted effort to be amongst the elite.”