Everton could move into 16th in the Premier League on Monday after defeats for Leeds United and Southampton on Sunday.
A double from Jefferson Lerma put Bournemouth on the front foot early on, although Leeds were able to get back in the game before the break through a Patrick Bamford goal.
However, the hosts struck again through Dominic Solanke and Antoine Semenyo to seize control of the game in the second half, eventually winning 4-1. While Bournemouth are now well clear of trouble at the bottom, Leeds remain entrenched in the battle to stay up after a testing spell under new manager Javi Gracia.