Hearts gained one of their best results in Europe thanks to a late goal from Mark De Vries in Bordeaux.
The Edinburgh side were under the cosh for most of the second half but the Dutch striker scored the decisive goal in the 79th minute, after a Kevin McKenna header had come off the post.
It was a tactical triumph for Hearts manager Craig Levein who raised a few eyebrows with his starting 11.
Levein left out experienced midfield player Scott Severin, and Australian Patrick Kisnorbo was given an unusual midfield role, but the switch paid dividends.
Hearts made their intentions clear from the start by playing five at the back and trying to pick out De Vries with long balls on the counter-attack.
The Jambos were disciplined and well organised in the first half and it was not until the 27th minute that Bordeaux had a shot at goal.
Indeed, Hearts could have had a penalty just before hand when Bruno Basto was adjudged to have fouled Robbie Neilson on the edge of the area.
It looked a soft decision but television evidence clearly showed that the challenge was inside the penalty box and, if the challenge was deemed illegal, then it should have been a spot-kick.
The French side stepped up a gear after that and Craig Gordon parried a David Jemmali shot from distance, before Andy Webster hooked the ball to safety.
Pascal Feindouno then shot narrowly over from 25 yards as Hearts were pushed deeper into their own area.
But Hearts weathered the storm and were almost gifted an opening goal six minutes from the break.
Neilson's cross was headed back towards his own goal by Feindouno and it took a fingertip save from Ulrich Rame to prevent the goal.
There was a lucky escape for Hearts at the start of the second half when Jean-Claude Darcheville controlled a diagonal pass from Franck Jurietti before rifling a shot just over the crossbar.
Darcheville came even closer in the 61st minute when his shot beat Gordon but struck the post. Andy Webster cleared the ball off the line and the goalkeeper smothered the follow-up shot from Paulo Da Costa.
Bordeaux threw everything at Hearts in the closing stages and it took a superb save from Gordon to keep out a netbound effort from Mauricio Pochettino with 14 minutes left.
Hearts snatched victory three minutes later when De Vries reacted first to touch the ball across the line after McKenna had seen his header hit woodwork.