The German side have only lost once so far in the group and will go through as group winners after Maccabi Tel-Aviv could only draw against APOEL in Nicosia.
The defeat means Bordeaux's poor European campaign will end following their final group game in Israel as they remain rooted to the bottom of the pool.
Cheick Diabate came close to opening the scoring for the hosts but his fifth-minute header was off target, whilst Sebastian Rode had Frankfurt's first decent chance with a header of his own.
Frankfurt's goalkeeper Kevin Trapp was tested for the first time midway through the first half when he kept out Henri Saivet's close-range effort with his legs.
Diabate wasted another good heading opportunity as the first half continued to ebb and flow, with both sides looking to force an opening.
Trapp was also equal to Carlos Henrique's 20-yard shot as the first half ended all square.
The hosts continued to press after the interval and Trapp was once again their main tormentor as he kept out Julien Faubert's shot.
Trapp maintained his fantastic showing with arguably his best save of the evening as he tipped Saivet's overhead kick wide of the post as the hour mark approached.
Frankfurt did not really test Bordeaux's goal until later in the game as Pirmin Schwegler's 73rd-minute effort flashed wide of the goal.
Lanig finally broke the deadlock with nine minutes remaining as he latched on to Tranquillo Barnetta's cross to convert from close range.
Despite being on the verge of elimination, Francis Gillot's side could not muster a response and were condemned to their fourth defeat in five Group F games.