However, Bournemouth were nonetheless able to maintain their unbeaten record at home in all competitions this season.
Bournemouth dominated the first half, moving the ball well at speed while Palace struggled to get a foothold but Roy Hodgson's side were a different proposition after the break.
Burnley's 4-0 win over Bournemouth was highlighted by Ashley Barnes's two goals.
Ryan Fraser's two goals led Bournemouth to a 4-2 win over Leicester City.
The result left Chelsea joint top with Liverpool on a maximum 12 points.
Bournemouth coach Eddie Howe has urged his team to seize the initiative in their future games, especially against the leading Premier League teams.
Eddie Howe said that the fact that his Bournemouth team have endured many positive and negative experiences alike has made them a better team.
Simon Francis is hoping to be in the starting line-up when Bournemouth play against West Ham.
Bournemouth have signed Jefferson Lerma from Levante.
He has now become Howe's second pre-season signing following the arrival of Wales midfielder David Brooks from Sheffield United earlier this month.
Bournemouth's difficulty in signing new players ahead of next season.
According to reports, Kane's injury is serious enough that Kane is facing a race against time to return to action in time for the World Cup in June.
Chelsea beat Norwich by way of a penalty shootout, while Bournemouth lost to Wigan.
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