The Cottagers' defeat - coupled with Sunderland winning 1-0 at Manchester United - was enough to send Magath's side down ahead of their final home game with Crystal Palace next week.
Fulham were poor from the off and goals from Peter Odemwingie, Marko Arnautovic, Oussama Assaidi and substitute Jon Walters sealed their fate, with Kieran Richardson scoring a consolation.
The relegation was Fulham's first since they plunged into the league's bottom division in 1994 - and also Magath's first in his lengthy career with nine different clubs.
He told Sky Sports: "I have to apologise, for not managing the situation and I was convinced that we were not going to be relegated.
"But today there was not enough for staying in the league.
"We were never in the game. For me, the team has felt too much pressure. They didn't act, they only reacted, and so you cannot win the game.
"It was the worst game since I have been here today because it was the only game that we had no chance from the beginning.
"For me, as I told you before, the team was not made for a relegation fight and so was the performance today."