The Cherries skipper elbowed Rooney in the box 31 minutes into the game, which left the Toffees forward with a bloodied eye, but the incident went unpunished.
Bournemouth went on to take the lead early in the second half as King lashed home with a low 20-yard finish which got the better of Jordan Pickford.
The Everton goalkeeper then had to be at his best to keep out a Jermain Defoe effort with his feet, as the home fans grew ever more restless with their unimpressive season so far.
There was more to come from the out-of-favour forward eight minutes from time, as a shot from Davies was blocked and the ball looped into the air for Niasse to poke into the net.
The result means that Everton end a run of three straight defeats in the league to move up to 13th in the table, while Eddie Howe's struggling charges stay 19th.