The Carling Cup holders, lacking several key players through injury and suspension, produced their worst performance of the season to leave their top-flight status on a knife edge.
Blues are now level on points with Blackpool and Wigan, in 18th and 19th, and have to visit Tottenham next Sunday. Blackpool are away to Manchester United while Wigan travel to Stoke.
Fulham boss Mark Hughes would have expected a response from his side after the 5-2 home defeat by Liverpool on Monday.
But he could not have imagined his players having such a stroll to victory and only poor finishing prevented them from at least doubling the final victory margin.
The Blues defence was at sixes and sevens and could not cope with Bobby Zamora, Andy Johnson and Clint Dempsey.
They were not allowed to settle in midfield and posed the minimum of threat up front without the pace and energy of the injured Cameron Jerome as an outlet.
To add to their woes Lee Bowyer, Stuart Parnaby and Martin Jiranek all had to be substituted through injury and then Alexander Hleb limped off to leave them playing with 10 men for the final 13 minutes.
Dempsey missed a great chance to put Fulham ahead after two minutes.
Former City striker Andy Johnson sent a low ball across the six-yard area and Dempsey spooned his shot over the bar with the goal at his mercy.
Blues looked nervous and Ben Foster had to parry aside a massive clearance from Mark Schwarzer at full stretch.
But from the resulting corner Brede Hangeland put the Cottagers in front.
The Birmingham defence was undone by Greening's corner and the unmarked Hangeland made no mistake from close range.
Fulham were threatening to score everytime they ventured forward and Andy Johnson glanced a header wide from Greening's cross.
Fulham came close to doubling their lead after half an hour when Zamora crashed a header against the post from a Greening free-kick.
City almost levelled after 34 minutes when Martin Jiranek's flick grazed the outside of a post from Parnaby's left-wing centre.
Phillips had his first chance after 41 minutes and his header was deflected wide from Larsson's first decent cross of the game.
Blues were back-pedalling from the start of the second period and Zamora missed two great chances.
He got on the end of an Aaron Hughes centre but directed his header straight at Foster from close range.
Then Zamora found himself with only Foster to beat but tried to chip him and did not test the former Manchester United player.
Fulham were in complete control and Sidwell crashed a low shot against the post with the home defence in disarray.
It came as no surprise when Hangeland made it 2-0 after 49 minutes.
Zamora's header was cleared off the line from Greening's corner but Hangeland pounced to tuck away the rebound.
Zamora picked up a knock and was replaced by Eidur Gudjohnsen, who was quick to test Foster with a low drive.
Andy Johnson, Carlos Salcido and Danny Murphy were booked in quick succession for dissent, time-wasting and fouling Bentley respectively.
Gudjohnsen became the fourth Fulham player to be yellow-carded for tripping Hleb, who then went off injured to leave his side a man short for the finale.