Henry Slade started the scoring with a penalty early on, before the first try arrived moments later when Tom Johnson and Jack Nowell combined to find Olly Woodburn, who slid over in the corner before Slade struck the post with the conversion attempt.
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Bordeaux responded well with a converted try in their first extended spell of possession as they shifted the ball out wide at pace, which allowed Nans Ducuing to cross over close to the touchline.
However, Exeter soon extended their advantage with a powerful driving move which ended with Moray Low scoring the try, before Slade sent over the conversion.
Space further up the field was harder to find as the first half progressed, but Slade continued his productive evening with two penalties to move Exeter 21-7 ahead at the break.
A vital third Exeter try arrived early in the second half when a patient charge up the field resulted in Ian Whitten evading a tackle for the try, but Slade was unable to add the conversion.
The door was back open to Bordeaux when Julien Rey burst through the line to bounce off a couple of tackles on the way to a try, before the conversion was missed by Lionel Beauxis.
Bordeaux showed their resilience once again as they responded to a second try from Woodburn by reducing the deficit through Yann Lesgourgues, who slide under the posts to complete a fine move from the French side.
The converted try reduced the deficit to 12, but Slade hammered another nail in the coffin with a penalty late on as Exeter claimed the win.