Joshua King scored with just one minute left to break Crystal Palace fans' hearts.
Luka Milivojevic, who has also been a key cog in Hogdson's side this season, gave the visitors the lead with a stunning free-kick for his tenth goal of the season and third in his last three.
Like they have so often this season, Bournemouth started to play after going a goal down but Eddie Howe still wanted to make a change. It proved to be a good decision.
He sent on Lys Mousset and the forward rewarded him with an equaliser three minutes later.