The 20-year-old had loan spells at Derby County and League One side MK Dons last term, scoring 25 goals in 53 appearances in all competitions.
Bamford, who began his career at Nottingham Forest, cost Chelsea in the region of £1.5m when he joined in 2012.
Boro boss Aitor Karanka confirmed his interest in him following a 1-0 defeat at Leeds earlier this month.
"Middlesbrough jumped out at me when I spoke to the manager and heard his aspirations," Bamford told the club website after sealing his move to the Riverside.
"When he put forward to me what he was looking to achieve, I was eager to come here and play."
Middlesbrough host Reading on Saturday.
Chelsea striker Patrick Bamford has joined Championship side Middlesbrough on loan until 1 January.