Romelu Lukaku now has 16 goals in the Premier League this season
The 23-year-old struck almost straight from kick-off and was gifted another – after a fortuitous second for James McCarthy – as the hosts raced into a 3-0 half-time lead.
However, a Josh King double set up a tense finish before Lukaku added two more late on, becoming Everton’s second-highest Premier League goalscorer – just one behind Duncan Ferguson’s 60.
And there was time for Harry Arter to tap in the visitors’ third ahead of Ross Barkley scoring Everton’s sixth.
Ross Barkley added Everton's sixth in stoppage time
Bournemouth haven't won in their last six games
“Romelu showed that he is one of the best finishers in football,” Everton manager Ronald Koeman said.
The Dutchman had hailed the “different spirit” within his Everton squad after a January clear-out of Darron Gibson, Bryan Oviedo, Tom Cleverley, Oumar Niasse and Gerard Deulofeu.
His new arrivals Morgan Schneiderlin and 19-year-old Ademola Lookman, handed his full Premier League debut, have brought a much-needed dynamism to his side that was clear for all to see yesterday.
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The team is gradually turning into one which Koeman believes should be targeting sixth-placed Manchester United – two points ahead albeit with a match in hand – in the quest for European honours this season.
Meanwhile, Cherries boss Eddie Howe admitted: “We tried to do something different today, but it didn’t work.”