Shane Long stooped low to nod home with just three minutes of normal time remaining, after Nikica Jelavic had given the visitors hope of a point at Craven Cottage.
Ashkan Dejagah showed his quality to put Fulham ahead when he produced a wonderful shimmy and strike 10 minutes into the second half.
Just three minutes later, Fulham looked home and dry when Amorebieta headed home his first goal for the club.
But Felix Magath could only look on in horror as Sascha Riether crashed a clearance against his own crossbar, allowing Jelavic to head home with 14 minutes to play.
And Hull showed commendable spirit to claw level as Long scored to all-but secure their top-flight status and provide a confidence boost ahead of next month's FA Cup final.
After John Arne Riise and Hugo Rodallega went close for Fulham early on, Jelavic had Hull's first sight of goal - David Stockdale comfortably saving the Croat's close-range header - though the opening half-hour boasted few clear-cut chances.
Indeed, the main talking point of the half came five minutes before the break when Long went down in the area under pressure from the tracking Amorebieta.
However, Lee Mason waved away Hull appeals, before the second-half began with an explosive Rodallega volley he was unable to direct on target.
Riise and Lewis Holtby had been replaced by Dejagah and Kieran Richardson at the break, with Magath forced to make his third change just eight minutes into the second half.
Darren Bent replaced the injured Steve Sidwell in an attempt to offer Fulham some attacking impetus but it was Dejagah who opened the scoring with a curled effort that left Steve Harper with no chance.
The Iranian cut inside David Meyler before whipping in his fifth league goal of the season and the hosts doubled their lead soon after through Amorebieta.
While the Venezuelan may have been fortunate to be on the pitch, Hull could have no complaints with the quality of the goal - the former Athletic Bilbao man expertly guiding a precise Richardson cross past Harper.
However, it all went wrong for the home side from then on in as Riether's unfortunate attempted clearance was tipped onto his own crossbar by Stockdale before Jelavic tidied up the rebound.
With Hull piling on the pressure - Sone Aluko and Ahmed Elmohamady both hitting the woodwork - the visitors' late surge eventually proved too much for Fulham to hold back when Long headed home the equaliser.
A point makes life a little easier for Hull, who are edging closer to mathematical safety, though Magath's men remained mired in the relegation zone after a stunning collapse.
