From the opening whistle the French champions, who qualified from the group stage with 16 points - the most of any team, looked too powerful for their Greek opponents, pinning them into their own half and barely relenting.
Where Laurent Blanc's side, who currently top the French League and have a French Cup final at the end of March, showed imagination and energy early on, Olympiakos lacked the necessary ambition until a frantic finish to the game.
The free-kick which led to the opening goal came from Vassilis Torosidis' innocuous challenge on Wendel.
And what followed was stunningly straightforward as, from wide on the left of the area, Gourcuff struck a wicked, curling effort beyond Nikopolidis.
And the impressive former AC Milan midfielder, who was a constant threat, almost doubled the lead when he latched on to Benoit Tremoulinas' infield ball but Nikopolidis reacted quickly to bravely make the save six yards out.
Gourcuff, who has 17 caps for France, was to get even closer when he stepped up to loop a 35-yard free kick over the head of Nikopolidis, but this time the goalkeeper was saved as the ball rattled off the bar and post.
After a quiet first half Derbyshire finally made an impact on the stoke of half-time as the former Blackburn man tapped the ball beyond Cedric Carrasso and into the back of the net only to see the referee's assistant flagging for offside.
Ironically, once Derbyshire was shown a second yellow card for a late challenge on Jaroslav Plasil in the second half after an earlier caution for dissent, Olympiakos improved considerably.
A throw-in from the left found Konstantinos Mitroglou unmarked and the substitute, who had only been on the field for two minutes, lashed his shot into the top corner to set up a tense final 20 minutes.
And the evening improved further for the Greek side when minutes after the score was returned to parity so too were the number of players on each side.
Deep in the Olympiakos half an inexplicable lunge from captain Alou Diarra saw the Frenchman handed a second yellow card.
Following the dismissal the previously slick Bordeaux looked rattled and were forced to cling on as the visitors pressed for another vital away goal.
But after Raul Bravo's shot was deflected wide for Olympiakos, Bordeaux finally killed off the tie.
With two minutes to go Tremoulinas' delivered a cross to the back post which found Morocco striker Chamakh, a player linked with several English clubs including Arsenal, and he leaped high to power his header beyond Nikopolidis.
For Bordeaux, who became the second French side to progress following Lyon, the win ensured a first quarter-final appearance since 1988.
Bordeaux midfielder Wendel:
"We scored the goal but then we shouldn't have taken any risks.
"They equalised when it was 10 against 11 and then we had a few problems.
"For the next round, I have no preference. There are great sides like Barcelona, Inter and Manchester (United) who are playing well at the moment.
"With these teams it would be 50-50. If it's Lyon it'll be difficult."
Olympiakos manager Bozidar Bandovic:
"We scored a goal and we had some chances, I congratulate my players.
"By believing in ourselves a bit more, perhaps things could have gone differently.
"I ask myself how we managed to concede those two goals but we looked our opponent in the eye."
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