Brugge exited last season's competition in the group stage but went on a streak of four successive games without defeat to end their participation in the tournament.
They had much the better of Wednesday's Group A encounter at the Jan Breydel Stadium but wasted too many opportunities in a game they deserved more from.
The stalemate does little for the hopes of two teams expected to be battling for the Europa League place in a group containing heavyweights Real Madrid and Paris Saint-Germain.
Brugge dominated from the off and should have led in the eighth minute when Emmanuel Bonaventure Dennis had the goal at his mercy but could only meet Ruud Vormer's cross with a woefully off-target header.
Matej Mitrovic then shouldered a free-kick over the crossbar and Brugge were almost made to pay for their profligacy, Gala's Ryan Babel unable to beat Simon Mignolet after a defensive mistake sent him through on goal.
There was more frustration for Brugge as Federico Ricca hit the crossbar on the half-volley at the end of a counter-attack after a pass deflected into his path, with Dennis then wasting another gilt-edged chance, collecting a long ball only to drag wide with just goalkeeper Fernando Muslera to beat.
Muslera came to Gala's rescue twice in quick succession, first to again deny Dennis before then keeping out David Okereke's header with a superb diving save.
Sofiane Feghouli clipped the top of the crossbar with a curling effort and Yuto Nagatomo fired wide from long range but Gala could not claim a victory they would not have merited.
What does it mean? Stalemate benefits nobody
In a group where the odds are stacked against them with Madrid and PSG expected to progress, the onus was on one of these clubs to earn a positive result to boost hopes of becoming a surprise package. Despite Brugge's best efforts, neither side did so and they each figure to be counting the cost of that failure come the next round of fixtures.
Muslera preserves point
Brugge would have had taken maximum points if not for the reflexes of Uruguay keeper Muslera, whose fine two saves early in the second half proved pivotal.
Dennis lacks menace
Nigerian striker Dennis has already found the net three times this season but he was guilty of spurning several chances to put Brugge ahead before being substituted in the second half. A better side would have punished his wastefulness.
Brugge must now turn their attention to the rather large matter of a crunch domestic clash with Anderlecht. Their next Champions League assignment is at the Santiago Bernabeu against Madrid. Gala visit Yeni Malatyaspor before a huge league match with Fenerbahce. They welcome PSG to Istanbul on October 1.