Both goals for Palace, who had no shots on target in their defeat to Fulham on Boxing Day, came from set pieces in the first half.
Jordan Ayew opened the scoring when he nodded in a corner from Michael Olise, who also set Eberechi Eze for the Eagles’ second.
New Bournemouth owners Bill Foley and actor Michael B. Jordan were shown onto the pitch before kick-off, but Hollywood’s sprinkling of stardust did little to help lift Gary O’Neil’s side.
There was just one change for Bournemouth, who launched Jaidon Anthony in place of Jack Stacey, while Patrick Vieira was forced to make two following James Tomkins and Tyrick Mitchell received red cards last time.
Marc Guehi, back from his own suspension, replaced Tomkins while Joel Ward replaced Mitchell.
Both teams paid their respects to Pele with a one-minute applause before action began in a heavy downpour.
Palace had a chance when the ball fell on Jeffrey Schlupp inside the box, but their effort was directed well beyond the left post.
The visitors were unable to do anything from a handful of early set pieces but took the lead after 19 minutes when Ayew rose high to meet Olise’s corner and cushioned a header in the bottom right corner from the edge of the six-yard box.
Olise looked to place another as he sent the ball across the face of goal from the right, but was ultimately unable to find a team-mate to place it past Mark Travers.
Palace then conceded their own free-kick, fired by Anthony who floated it down the pitch to Kieffer Moore on the left. The Cherries striker headed a threatening header into the box, forcing Vincente Guaita into a leaping hold to avoid any danger of an equaliser.
The Eagles extended their lead after 36 minutes after Olise again placed a team-mate from a perfectly placed corner.
This time it was Eze, who wasted no time curling a right footed effort into the bottom left corner close to the penalty spot.
It was the second time this season Palace had entered the second half with a two-goal cushion, and the first time both had come from Palace players, with Manchester City’s John Stones own goal helping the Eagles in August .
The rain died down after the restart, as did the home support, whose team seemed calmer. Guaita somehow survived several attempts by Bournemouth in a scramble before Travers blocked Eze’s spirited effort directly from a free-kick at the other end.
The hosts ultimately could only take consolation that the deficit was not wider after Schlupp selected Jean-Philippe Mateta, but the substitute headed his effort wide.
Guaita stopped Siriki Dembele’s strike before Wilfried Zaha skied in a late effort and the visitors pushed back four minutes of added time to seal the clear victory.
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