Magath has been handed an 18-month contract and replaces head coach Rene Meulensteen, who has been relieved of his duties.
After learning of his sacking Meulensteen told BBC Sport: "They have hit the panic button on emotion and fear - but hey ho, that's football."
In an open letter on Facebook, Magath said he had chosen Fulham because it presented him with a "fantastic working environment."
"This traditional, great club and its owner Shahid Khan have completely convinced me to take the job of manager," he said.
As a manager, Magath has won three Bundesliga titles - two with Bayern Munich (2005 and 2006) and one with Wolfsburg (2009).
The 60-year-old has spent his entire career as a player and manager in the Germany and has been out of work since leaving a second spell in charge of Wolfsburg in 2012 by mutual consent.
"Felix is an accomplished manager with multiple honours in the Bundesliga and a hunger to replicate his success with Fulham in the Premier League," said chairman Shahid Khan.
"I’m especially impressed with the reputation Felix has for coming into clubs at difficult times, often late in the season, and lifting them to their potential and beyond. Felix knows that is precisely the task awaiting him at Fulham, and he made it abundantly clear that he wants and is ready for the opportunity."
"Our club has shown promise in recent matches but the fact is we have not won a league match since 1 January. Given our form, we can no longer merely hope that our fortunes will finally turn. And with 12 matches remaining and at least four points separating us from safety, we certainly can no longer post empty results. Action was required."
The club's chief executive Alistair Mackintosh added: "The opportunity to bring in a manager with the experience of Felix Magath would typically be unlikely at this point in the season.
"However, Felix relishes this challenge. He is very impressed with our squad and commitment by Mr Khan and his confidence is precisely what we need over these final three months of the season."
Eurosport Germany's Dirk Adam gave us his thoughts on the shock appointment:
"The news that Felix Magath has been appointed Fulham manager is a sensation in Germany. Most people expected him to be named the new coach of Hamburg. At Hamburg though the situation is pure chaos so Magath has used this opportunity to surprise everybody although some people think that Fulham is not the right level for him.
He is a brilliant manager though. At Wolfsburg he was the coach AND the club's business manager rolled into one.
Magath's training methods in particular are legendary. He loves to put his players through strong conditioning. He trains with medicine balls and he likes to order his players up flights of stairs too.
As for his style of play, he loves to use a playmaker like Diego who had at Wolfsburg. He also prefers to play with two strikers up front to attack the opponent but attaches importance to a strong defence. Every part of the team is important to him. "