The 31-year-old has penned a three-year deal with the newly promoted Premier League side after impressing during his time with Palace.
Murray revealed his delight after completing the deal as he looks to register more goals in his bids to help his new club remain in the English top flight.
“It feels great to finally be here. The way the boys play football is very exciting. It’s front-footed and it will be nice to try and get some goals from it,” he said.
He managed to score 47 goals from a total of 125 appearances for Palace, a huge factor in their push for promotion during their time in the Championship.
The Cherries also completed a half-season loan for Joe Bennett from Aston Villa along with a season-long loan for River Plate ace Tomas Andrade.
However, the club suffered an injury blow after £7million summer signing Max Gradel tore his cruciate ligament and could now be unavailable for an extended period.