The former England Under-21 winger passed a medical and has signed a contract until June 2014.
No fee has been announced but it has been reported Middlesbrough will get around £7million.
Roberto Mancini: "We are delighted that Adam has chosen to come here.
"He is a very good player with fantastic potential, and as a club we are always keen to sign British talent."
Adam Johnson: "This is a dream move for me ... a once in a lifetime opportunity I could not turn down.
"Boro have been great to me. I got on well with Gordon Strachan, and he knows what this move means to me."