The champions took an early lead through Argentine midfielder Lucho Gonzalez but then had midfielder Edouard Cisse sent off after 34 minutes.
Bordeaux and Marseille now have four points from as many games, eight behind early leaders Toulouse who won 2-0 at Nancy on Saturday.
Marseille's new signing Loic Remy, who was cleared to play despite the discovery of a heart defect last week, had a penalty denied early on.
The striker went down in the box and TV replays clearly showed Bordeaux defender Mathieu Chalme tripped him up.
Marseille eventually took a deserved lead when Lucho tapped in a Brandao cross in the 12th minute.
Cisse was then shown a red card for a tackle from behind on Czech midfielder Jaroslav Plasil.
Goalkeeper Cedric Carrasso made two fine saves to prevent Bordeaux going two goals down, first from a deflected free-kick and then from a header by forward Andre Ayew.
Remy also hit the post for Marseille but Bordeaux's effort eventually paid off when Modeste headed in a Plasil cross one minute from time.
Elsewhere, Paris Saint-Germain crashed 3-1 at Sochaux.
"We played a pathetic game. Such a performance is unworthy of PSG," chairman Robin Leproux told the club's website. "We can't go on like this."
PSG trailed 3-0 at half-time before France striker Guillaume Hoarau pulled one back with a left-foot drive just after the break.
The visitors finished with 10 men after substitute Jeremy Clement was sent off in the 89th minute.
PSG have four points while Sochaux are in fourth spot on seven points.
Lille were held 1-1 at home by Nice and Auxerre were beaten 2-0 at Monaco.
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