The 10th-placed Cherries have amassed 48 points, two more than their previous best total of 46 achieved in the 2016-17 season. South-coast rivals Brighton, who finished sixth last year, are 12th on 44 points.
Brighton dominated possession in the first half but went behind when Senesi reacted quickest to head home from a corner in the 13th minute, and Turkish striker Unal added the second with a powerful looping effort seven minutes after the break.
Brighton substitute Julio Enciso had a number of good opportunities to pull a goal back for the visitors, but it was Kluivert who rounded off the scoring with a superb run and shot as the visitors' winless league run extended to six games.
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