Edinburgh were pegged back late on as Bordeaux snatched a draw that may prove crucial in Challenge Cup Pool 3.
Both sides had opened their campaign with victories last week and they were evenly matched at Murrayfield.
Blair Kinghorn's try was the highlight of a stuffy first-half, with Jaco van der Walt's boot helping the home side into a 13-6 half-time lead.
Semi Radrada responded after the break and Matthieu Jalibert's late penalty snatched a draw for the French side.
Edinburgh's second-string did the job in the pool opener away to Agen last week by delivering maximum points, but the home side's team selection gave credence to the idea this was a proper European night in Edinburgh.
Richard Cockerill made 13 changes, recalling a host of front-liners, including talismanic number eight Bill Mata for his first appearance of the season.
Bordeaux arrived off the back off an excellent start to their domestic league season, sitting second in the Top 14 after six wins and a draw from their nine games.
Both packs looked strong, there were threats galore in both back-lines and yet the opening half-hour was lacking in much meaningful action at either try-line.
Two Van der Walt penalties either side of a three-pointer from Bordeaux fly-half Jalibert established a 6-3 lead for the home side, but this was a match in dire need of a try and, when it arrived, it was a beauty.
From turnover ball, Edinburgh launched a counter-attack. Grant Gilchrist recognised the overlap out wide and shipped the ball to Van der Walt. Matt Scott straightened the running line and put Kinghorn into space and the full-back showed great gas to outstrip the cover defence and slide over in the corner.
Some Edinburgh ill-discipline at the breakdown allowed Jalibert to reduce the arrears, but the boot of Van der Walt re-established the 10-point lead early in the second half.
Just when the home side looked to be getting a grip on things, Bordeaux struck with a scintillating try.
Edinburgh lost the ball in contact and a succession of quick offloads took the visitors into the danger area. Replacement prop Ravai Kovekalou sliced through the midfield defence in a fashion Semi Radrada would be proud of before the electric Fijian centre himself took over and danced away from a couple of tackles on his way to the line.
After the stodginess of the first half, the game was now breaking up and that seemed to play into the hands of the French side, with Radrada and Santiago Cordero starting to find holes.
Jalibert pinged a long-range penalty over to level things up at 16-16 in the dying minutes. Neither side could fashion an opportunity to win it in the final moments and Bordeaux may well be the more satisfied with the result given the teams are due to meet in France later in the pool.
Edinburgh: Kinghorn, Sau, Bennett, Scott, Van der Merwe, Van der Walt, Pyrgos; Schoeman, Willemse, Ceccarelli, Toolis, Gilchrist, Bradbury, Crosbie, Mata.
Replacements: Fenton, Bhatti, Berghan, Douglas, Barclay, Shiel, Hickey, Taylor.
Bordeaux: Buros, Cordero, Radradra, Seuteni, Connor, Jalibert, Lucu; Paiva, Pelissie, Kaulashvili, Douglas, Flanquart, Gorgadze, Woki, Higginbotham.
Replacements: Kovekalou, El Fakir, Marais, Tauleigne, Gimbert, Botica, Tamanivalu.
'We've got to be smarter' - reaction
Edinburgh head coach Richard Cockerill told BBC Sportsound: "Even though it is the second tier of Europe, they are a good side.
"I am disappointed we did not win the game and we had enough of the game to do that.
"We need to learn to make the right decisions at the right time and good sides always get the chance to score and they did that.
"It is small margins, we were as good as they were and should have won the game. We've got to be smarter and learn our lessons quickly."