Blues, 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.
They are now level on points with Blackpool and Wigan, in 18th and 19th, and have to visit Tottenham next Sunday. The Tangerines are away to Manchester United while Wigan travel to Stoke.
Clint Dempsey missed a great chance to put Fulham ahead after two minutes. Former Blues 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 the Cottagers went front.
The Blues defence was undone by Jonathan Greening's corner and the unmarked Hangeland made no mistake from close range.
Fulham came close to doubling their lead after half an hour when Zamora crashed a header against the post from a Greening free-kick.
Blues almost levelled after 34 minutes when Martin Jiranek's flick grazed the outside of a post from Stuart Parnaby's left-wing centre.
Fulham were in complete control. Sidwell crashed a low shot against the post with the home defence in disarray and 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.