The Spanish midfielder saw his penalty saved by Gabor Kiraly but followed up to put away the rebound.
The defeat leaves Palace 11 points adrift of second place Sheffield United and it now looks as if Iain Dowie's team must plan for the play-offs.
It had looked so promising when Clinton Morrison scored in the 80th minute, but Derby responded with a superb goal from Idiakez 10 minutes later and both teams could have won the game in the second half.
But both goalkeepers were inspired and the match appeared to be heading for a draw until Darren Ward tripped up Kevin Lisbie right at the end.
Palace will look back and wonder how they lost after being in front at Derby for the second time in three seasons.
They started positively and it was no surprise when Morrison scored after Lee Camp had parried a fierce shot from Andy Johnson.
But Derby showed their spirit by forcing Palace back and Idiakez equalised when he cut in from the left before curling a 20-yard shot into the top corner.
Jobi McAnuff struck the crossbar from 25 yards before half-time and after the break it was a case of missed opportunities at both ends.
Ben Watson was twice denied by Camp in the 46th minute and then saw his third attempt cleared off the line by Richard Jackson before Johnson had a fierce right-foot shot pushed away by the keeper.
Camp had to tip over a 25-yard drive from Aki Riihilahti in the 59th minute and Palace were convinced they should have had a penalty when Camp appeared to trip Johnson after he had spilled a shot from Watson.
But Derby had chances as well and Giles Barnes should have done better when he was put clear only to over run the ball.
Idiakez also wasted a one-on-one with Kiraly in the 83rd minute and the goalkeeper won another personal battle with Lisbie when Adam Bolder put him through.
The match seemed to be heading for stalemate until Barnes set up Lisbie with a back header and Ward brought him down to concede a penalty.
Kiraly went the right way to beat out Idiakez's penalty but the ball fell kindly for the midfielder to drive the rebound into the net.