Jon Dagur Thorsteinsson put Fulham in front three minutes after half-time, before Jayden Harris sealed the visitors' victory at Rockliffe Park.
Boro are behind leaders Aston Villa on goal difference. Fulham move up to fourth, one point and two places adrift of Boro, having played one match more than the top three.
Boro: 29 Jan v Aston Villa (A) PL2
Fulham: 29 Jan v West Brom (A) PL2