A third successive defeat leaves Groves' men with just one win in 12 league and cup games this season.
Midfielder Hope Akpan gave Crawley the lead after 31 minutes with his third goal of the season, firing home from 12 yards after a low cross by Nicky Adams was cleared off the line by Stephane Zubar.
Former Swansea defender Joe Walsh doubled the advantage on the stroke of half-time by heading in a free-kick from Mat Sadler from six yards,
Lee Barnard struck for Bournemouth seven minutes after the break with a close-range left-footed shot after a cross from the left by Charlie Daniels struck Mark Connolly.
But Adams restored Crawley's two-goal cushion shortly after the hour mark when he seized on to a pass from Nicky Ajose to beat former England 'keeper David James. James then denied Crawley further goals by tipping over a close-range header from Ajose before superbly keeping out a dipping free-kick from Adams.