Edmonton got on the board first at 4:27 of the second period when Fernando Pisani scored on a rebound from near the left post.
Jeff Deslauriers gave up the four goals on 39 shots for the Oilers, who lost for the eighth time in their last 11 games.
Cullen scored a power-play goal from the right circle at the 8:24 of the third for what proved the game-winner.
Edmonton failed to score on four chances with an extra skater.
Ottawa has beaten Edmonton six straight times.
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