Headers by Frank Lampard and Joe Cole, in his 50th European match for the club, put Chelsea in control after 30 minutes and only wayward shooting from Lampard and Nicolas Anelka before the break, prevented the London side winning by a bigger margin.
The goals in the final 10 minutes from Florent Malouda and Anelka gave the scoreline a far more realistic look as Chelsea dominated the game from the first kick to the last.
France striker Anelka slotted in his first Champions League goal for Chelsea from point blank range with the last kick of the match after a 30-metre shot from substitute Juliano Belletti was palmed on to the bar by Bordeaux goalkeeper Ulrich Rame.
Minutes earlier Anelka had fired against the post when it looked easier to score.
Chelsea coach Luiz Felipe Scolari, taking charge of a team in the Champions League for the first time, only had to wait 14 minutes to see his first goal when the unmarked Lampard headed in a simple chance from Jose Bosingwa's cross from the right.
It was only a matter of time before Bordeaux's shambolic defending and Chelsea's speed of passing and movement on and off the ball brought more goals and the Premier League leaders duly doubled their lead after half-an-hour.
Cole reacted first to Lampard's inswinging corner from the left and outjumped the much taller defenders Franck Jurietti and Fernando to leave goalkeeper Rame well beaten again.
The Bordeaux captain had saved his side eight minutes earlier though when he somehow got a hand to the ball and grasped it before it crossed the line after Fernando almost turned a Bosingwa shot into his own net.
The French side only seriously threatened twice in the match, firstly after only 10 minutes when midfielder Wendel lashed a 25-metre shot just over Petr Cech's bar after a misplaced clearance from John Obi Mikel.
The Brazilian also fired a free kick over the bar although Cech always looked to have the shot covered.
Chelsea were so much in control that little of note happened after the break until Malouda brightened things up in the 82nd minute when he made it 3-0 following fine approach work by Jon Obi Mikel and a Lampard backheel that set up the French winger.
Anelka rounded off a comfortable night for Scolari's team by scoring with the last kick of the game.
The Bordeaux captain had saved his side eight minutes earlier though when he somehow got a hand to the ball and grasped it before it crossed the line after Fernando almost turned a Bosingwa shot into his own net.
The French side only seriously threatened twice in the match, firstly after only 10 minutes when midfielder Wendel lashed a 25-metre shot just over Petr Cech's bar after a misplaced clearance from John Obi Mikel.
The Brazilian also fired a free kick over the bar although Cech always looked to have the shot covered.
Chelsea were so much in control that little of note happened after the break until Malouda brightened things up in the 82nd minute when he made it 3-0 following fine approach work by Jon Obi Mikel and a Lampard backheel that set up the French winger.
Anelka rounded off a comfortable night for Scolari's team by scoring with the last kick of the game.