They had come back from a goal down to take the lead with 12 minutes left, but 90 seconds later Derby were level and the Londoners have now failed to win away for 18 months.
If they do go down, they may look back on this game with regret because they should have held on after a deflected shot from Hameur Bouazza beat Roy Carroll.
But Derby have a striker in Villa who could blossom next season and he made no mistake when his header beat Kasey Keller, who had made some great saves to keep his team in the game.
He could do nothing about Derby's 10th minute opener which owed everything to Villa's instincts in reacting sharply to turn in a shot from Dean Leacock that was going wide.
But Leacock was at fault at the other end in the 24th minute when he and Andy Todd allowed Diomansy Kamara to get between them and head in Simon Davies' cross from the right-hand edge of the box.
Kenny Miller was denied by a brave save from Keller two minutes later, but went off injured before half-time to be replaced by Rob Earnshaw.
Both teams needed to win and the second half was full of near misses as Jimmy Bullard struck the top of the bar with a 25-yard free-kick, while Earnshaw flashed a shot just wide after robbing Keller on the left-hand edge of the area.
Robbie Savage should have restored Derby's lead when Villa's header set him up, but he shot straight at Keller from 12 yards and, seven minutes later, Fulham were in front.
Kamara looked offside as he broke clear on the left and, although his shot was parried by Carroll, Bouazza's follow up would have gone in but it took a big deflection off Leacock before hitting the net.
The celebrations were short lived as Mile Sterjovski crossed from the right and Villa rose to head powerfully past Keller.
Fulham thought they should have been awarded a penalty late on when substitute Eddie Johnson appeared to be tripped, but the points should have been safely locked away by then.
Derby have been prepared for relegation for some time but at least Villa and Sterjovski showed they have the talent to thrive in the Championship.
Fulham now look favourites to join them and it will take a dramatic change in form and fortune for them to stay in the top-flight.