Idiakez brilliantly curled in his eighth goal of the season from 25 yards in the first half while Wotton skied his second-half spot kick high over the crossbar.
To be fair, the Plymouth skipper slipped as he was about to strike the ball and that summed up the visitors' luck on an afternoon when Derby had to fight every inch of the way.
Lee Camp also made two stunning saves to deny Plymouth a share of the spoils and once again Derby were far from convincing in their own stadium.
Tom Huddlestone had one of his unhappiest games in the centre of defence and Plymouth will look back and wonder how they left empty handed.
Derby had started so brightly and were virtually camped in the Plymouth half for the first half hour.
Morten Bisgaard and Grzegorz Rasiak both missed chances before Peter Gilbert tripped Marcus Tudgay and Idiakez swept the ball over the wall and into the top right corner.
Derby certainly looked good going forward but they remained vulnerable at the back and Nick Chadwick should have done better when a Huddlestone error let him through but he dragged a shot wide.
Idiakez struck a post at the start of the second half but Rasiak then pushed over Graham Coughlan to give Plymouth a penalty which Wotton made a hash of.
After that Plymouth created the better chances and Camp needed to be at his most alert to keep out a header from David Norris in the 63rd minute.
Three minutes later he did even better when Tony Capaldi connected sweetly with a 25-yard volley that was heading for the top left-hand corner until the England under-21 goalkeeper clawed it away.