The 29-year-old has signed a four-year contract, with Reds reserve team players Lauri Dalla Valle and Alex Kacaniklic moving in the opposite direction.
Konchesky was at Anfield to watch his new team's victory over West Brom on Sunday but saw his deal was held up by negotiations and medicals at Fulham's end.
The left back, who played under Hodgson at Craven Cottage, will provide much-needed competition for Fabio Aurelio at left-back.
Argentinian Emiliano Insua is set to leave the club, with a proposed loan to Galatasaray, after an earlier move to Fiorentina collapsed over personal term.
Konchesky meanwhile has been allocated the No3 shirt at his new club.
Hodgson is repeating Rafa's mistakes...............signing many players that can make NO difference..........
this is the beginning of the end for Liverpool...................they can only go south until they find a super-rich buyer !