The 36-year-old ex-Belgium international dedicated the 4-2 victory at the Vitality Stadium to the fans as around 1,000 of them travelled to the South Coast.
Manuel Benson and Anass Zaroury grabbed two goals apiece to put the Clarets in the hat for the fourth round.
Kompany said: “The first mention is for our fans, it was a long journey and it is a result that I hope they can be proud about.”
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Kompany said: “The first mention is for our fans, it was a long journey and it is a result that I hope they can be proud about.”