"We had five or six good chances but we just didn't score the goals," said Luxemburgo.
"We kept putting it into the middle but we couldn't get it in the net. If you don't do that then an isolated play or a set piece can cause you problems.
"That is exactly what happened in the second half and that's when the nerves set in.
"If we'd have scored before then the game would have been over, but we were lucky enough to get the goal in the end."
Real looked to be cruising towards a comfortable win after Raul's record 50th European Cup goal. The Spain striker got on the end of a perfectly flighted cross from Beckham to head home at the near post in the ninth minute.
But the nine-times European champions, missing leading striker Ronaldo through suspension, spurned a hatful of chances to extend their lead and none of their forwards took advantage of a constant supply of scoring passes from Beckham.
Real paid for their profligacy when Olympiakos snatched an equaliser at the start of the second half, midfielder Pantelis Kafes scoring with a stunning 25-metre strike.
It was not until four minutes from time that substitute Roberto Soldado managed to grab the winner when he headed home a Beckham cross at the second attempt.
NERVES SET IN
"Beckham has been playing very well for a while, this isn't the first good match he has had for us," said Luxemburgo. "The win will give us more confidence and I'm still hoping of finishing as group winners."
Real lost their opening game 3-0 at Olympique Lyon.
For Olympiakos a second defeat in a row dealt a heavy blow to their hopes of qualifying for the knockout stage. Coach Trond Sollied said he was proud of his players' efforts.
"I think no one expected us to win in the Bernabeu, but my players can now see that it is possible to win if you have a good day. Every player did his best in the circumstances.
"Psychologically it would have meant a lot for us (to get a draw). But we know that when we are playing a stronger team we cannot afford to lose concentration for one minute.
"Our tactic was very simple," Sollied said. "We wanted to be so close on them that we should deny crosses from both sides. We succeeded with Roberto Carlos but not with Beckham."
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