Vincenzo Iaquinta had given the hosts the lead on 63 minutes when he fired past substitute goalkeeper Ulrich Rame after Fabio Cannavaro's ball forward outfoxed Bordeaux's defence.
Midfielder Plasil then poked in at the far post on 75 minutes after a shaky Juve defence had allowed a free kick to be flicked on.
The home side, in Ciro Ferrara's first European game as coach, were largely flat for much of the contest despite having won their first three Serie A games.
The French champions had just as many chances with Fernando slicing through Juve's backline early on and forcing Gianluigi Buffon into a smart save before again being denied by the Italy goalkeeper in the second period.
Claudio Marchisio rattled the bar for Juve late on but they lacked creativity without the injured Diego and Alessandro Del Piero.
"I'm sorry. We strongly wanted to start with a win but we came up against a well-organised side. The Champions League is always difficult but there is time to put it right," Ferrara told a news conference.
"Buffon has been phenomenal on a couple of occasions. I am a bit worried but it is normal. I don't want to find excuses."
Bordeaux coach Laurent Blanc, who media linked with the Juventus job before Ferrara was appointed in May, urged his side forward knowing Juve were not at their best but former AC Milan midfielder Yoann Gourcuff spurned a few chances.
"We worried Juventus and could have won," Blanc said." I'm happy with the result as before the match I would have taken a draw. What gives me more pleasure is how we played."
Before scoring, Iaquinta blazed wide from a tight angle after colliding with goalkeeper Cedric Carrasso, who was substituted, and former Olympique Lyon defender Fabio Grosso soon troubled new keeper Rame with a curling freekick.
A small group of Juve fans made racist noises towards Bordeaux centre back Michael Ciani after an on-field incident but the insults quickly stopped. Juve had to play one Serie A game behind closed doors last season following racist chanting.
