James McCarthy made it two 24 minutes in when he found the net from close range and Lukaku got his second five minutes later.
Ronald Koeman's side looked home and dry until Joshua King reacted quickly when the ball came to him in the box to grab a 59th-minute goal.
And he bagged another 11 minutes later as it appeared a comeback may be on.
But Lukaku then scored two more in the space of three minutes from the 83rd to run out winners before Harry Arter's late consolation.
Ross Barkley got Everton's sixth when he rounded Artur Boruc in the 95th minute.
STATS, FACTS, GOALS & LOLS
ROMELU LUKAKU equalled the record for the fastest Prem goal of the season – held by Chelsea’s Pedro at 30 seconds – as well as scoring the fastest-ever Prem goal at Goodison.
RONALD KOEMAN celebrated his 100th Prem game as a manager – and his 24th in charge of Everton going for a fourth win in five.
EDDIE HOWE brought in Sir Clive Woodward in midweek to give a motivational speech – yet the Cherries started like most of them were asleep.
FAIR PLAY to big contingent of Cherries fans who made the long trip north and the way they out – sung the Toffees support even after Lukaku’s stunning 14th of the season.
NO WONDER Eddie Howe looked disgusted as James McCarthy scored Everton’s second – he shanked his first attempt but worse the Cherries defence lay down against Romelu Lukaku’s burst from the right.
EVERTON’S THIRD goal from Romelu Lukaku meant the big Belgian has now scored 15 or more goals for five seasons in a row since first leaving Chelsea.
JAMES McCARTHY’S goal meant that for the first time in his career he had scored and assisted – it was his pass from which Lukaku scored the opener.
ADEMOLA LOOKMAN made his full debut and the 19yr old from Charlton continues to look the best 10M quid’s worth of talent in the January window.
JOSH KING scored a cute double inside 11 minutes that made you think what might have been for him had Manchester United not dumped him in the summer OF 2013.
ROSS BARKLEY’S back heel for Romelu Lukaku’s fourth goal meant he had created 49 chanes this season – the best from an English player.