Rovers were first linked with a move for Dartford-born Sheaf - whose younger brother Max is on the books of Hull City - in January and have maintained their interest in the player, who can also operate in central midfield.
Doncaster are understood to have beaten off interest from rival EFL clubs to land Sheaf, who played for the whole of last season with the Gunners' under 23s.
Prior to that, Sheaf - capped by England at under-18 and under-16 level - spent the second half of the 2017-18 season on loan at League Two outfit Stevenage.
On signing for Rovers, Sheaf said: "I have played at centre-back in the past couple of seasons, but I was in midfield prior to that - but I enjoy either and will do my best.
"I like to get on the ball and make things happen. I am really excited and looking forward to getting going now."
Sheaf - who joined the club from West Ham's academy in 2014 - has made two appearances for Arsenal's first team and made his debut as a substitute in the Europa League win at Red Star Belgrade in 2017-18.
The 6ft 1in versatile midfielder is Rovers' fourth signing of the summer following the additions of full-backs Brad Halliday and Reece James and midfielder Madger Gomes.