The 21-year-old, a product of the club's academy, cut in from the right flank on 72 minutes past defender Richard Jackson before curling a superb left-foot drive round Rams' keeper Lee Camp into the net.
"It's a dream come true, I've been waiting since I was 16 to get to this point and now I can grab it with both hands. It was a great feeling and I really enjoyed it. I just cut inside and hit it with my left peg," said match-winner Richardson.
The Whites should have doubled the lead straight away when Michael Duberry, who won nearly everything in the air, headed a free-kick back into the danger area and skipper Paul Butler, who had a brilliant debut, acrobatically blazed over the crossbar.
Leeds just about deserved the win after creating the better chances with Julian Joachim, one of seven debutants in the Leeds side, twice wasting golden opportunities to open his account.
The former Aston Villa and Coventry striker's best chance came in the 20th minute when a long ball forward was flicked into his path by Michael Ricketts from a Simon Walton clearance.
Joachim then raced clear of the defence only to pull his shot wide from 15-yards. He was also played clear early in the second half only for the outstanding Camp to race from his line and smother the ball.
Leeds were the better side in the first half but failed to take advantage of several good openings. The lively Tommy Smith went close for Derby at the other end.
Then Danny Pugh brought a brilliant relfex save from Camp with a header from an inviting Gary Kelly cross.
Ricketts and Pugh both failed to apply the finishing touch to Kelly's cross and then Camp had to be alert to keep out a dipping Pugh free-kick.
Derby's best effort came after just ten minutes when Morten Bisgaard hit a low shot just wide from 25-yards out.
Derby were unlucky not to take the lead on the stroke of half-time when Michael Johnson headed in following a right-wing corner, but the goal was disallowed after the assistant ref spotted that Idiakez's kick had drifted over.
The Rams were much improved after the break and were twice denied by Leeds keeper Neil Sullivan in the opening stages. First, the Scottish international bravely raced from his line to block a shot by Marcus Tudgay.
And then on the hour Sullivan had to be alert to finger-tip a Junior shot round the post after good build-up work by Derby.
Then Sullivan, in the dying moments, somehow flicked Junior's effort over the crossbar after Ian Taylor's looping header had come back off the post.
The visitors were then made to pay for that when Richardson opened the scoring and set Leeds up for 3 points, ensuring the club's proud record of having never lost on the opening day of the season since 1989 continues.