They will join fellow Scot Rory Lawson at Kingsholm after signing one-year contracts.
First news of the planned double move emerged last month, with Dickinson and Strokosch joining Edinburgh team-mates Simon Taylor and Rob Dewey in finding new clubs.
Taylor will join Stade Francais, while Dewey is moving to Ulster.
Gloucester head coach Dean Ryan was delighted to sign Dickinson and Strokosch, who are both expected to break into the senior Scotland side.
Ryan said: "We are careful who we want to attract but feel both of these players will add tremendously to the pool of talent we already have available.
"They [Edinburgh] played against us in the Heineken Cup this season and impressed us greatly. We are attempting to build a young, talented and competitive squad, and both of these players typify what we are looking for."
Dickinson and Strokosch both predicted the move would have a positive impact on their performance.
Scotland A prop Dickinson, 23, said: "Once I spoke to the coaches I was in no doubt that if I want to reach the highest levels then Gloucester offers the tough environment that can help me to achieve my goals."