Naysmith, 28, moves in the opposite direction to Phil Jagielka, who joined the Merseysiders from United on Wednesday for £4 million.
Naysmith said: "Coming to play under Bryan Robson makes me really excited and, after speaking with the chairman, I am keen and happy to be here. I was always there or thereabouts at Everton, but they had several established players and, if you were out injured, it was difficult to force your way back in."
Naysmith also revealed the new management team of Robson and Brian Kidd at Bramall Lane played a big part in his decision to swap the Premiership for the Championship.
"I spoke to Phil Neville [the Everton captain] about working with Brian Kidd and he had nothing but praise for his training methods and he was sure I would enjoy it immensely. Both Robson and Kidd are born winners.
"I have no problems with dropping a division, as I feel we will only be here for one year. We have a great squad with plenty of experience, so we should be aiming to get promoted at the first attempt."