The experienced forward has signed a three-year contract with the Bundesliga outfit, bringing to an end his prolific spell in the Primeira Liga.
Dost scored 81 goals in 103 appearances for Sporting across all competitions, establishing him as one of the most deadly players in European football.
His goals helped Marcel Keizer's side secure a domestic cup double last term, while also finishing third in the final league standings.
Frankfurt have been on the lookout for a new frontman since offloading talismanic duo Luka Jovic and Sebastien Haller, to Real Madrid and West Ham respectively.
Dost will now be expected to lead the line for Adi Hutter's team this season and his past experience in Germany suggests he could prove a shrewd acquisition for the club.
Sporting, meanwhile, left Dost out of their last two matchday squads, securing wins over Sporting Braga and Portimonense.
The Portuguese giants will be hoping that Dost is the last high-profile star to leave Jose Alvalade Stadium, with the final September 2 transfer deadline edging ever closer.
With the English transfer window now closed the 24-year-old looks set to stay put, but Real Madrid are also reported to be considering a late swoop for his signature.