Lukaku curled home an exquisite opener after just 30 seconds to equal the record for the fastest league goal of the season set by Chelsea's Pedro against Manchester United.
Everton vs Bournemouth 6-3 All Goals & HighlightsLukaku's trickery set up James McCarthy for Everton's second before the Belgium striker doubled his tally when Simon Francis' suicidal back pass gifted him the opportunity to loft the ball over Artur Boruc for a 3-0 half-time lead.
Joshua King's brace gave Bournemouth hope but Lukaku added two more inside the final 10 minutes to put the game to bed, although there was still time for Harry Arter to add a third for the visitors and Ross Barkley to slot in Everton's sixth.
The result was a fitting way for Everton manager Ronald Koeman to celebrate his 100th Premier League game, and takes the seventh-placed Toffees to within two points of Manchester United.
For slumping Bournemouth, meanwhile, defeat drops them to within six points of the relegation zone.
Eddie Howe's men - winless in 2017 - are now on a run of one victory in eight and have conceded 34 goals in their last 13 games - shipping at least three goals in nine of those.
Brilliant Lukaku steals the show
Everton's vital fourth goal came following a 21-pass move, which culminated with Coleman's pinpoint cross for Lukaku to complete his hat-trick.
"We said we wanted to come out flying, we worked on it all week," Lukaku said as he held the match ball at full-time. "There's no lack of confidence."
There certainly wasn't.
Everton completely bullied the opening period, helped of course by taking such an early lead when Morgan Schneiderlin broke up play in midfield and Lukaku exchanged passes with McCarthy before stroking a lovely shot beyond Boruc.
Lukaku then breezed past two markers to set up McCarthy to make it 2-0 - the midfielder scoring at the second attempt - before adding his second of the contest with a cool finish when Francis inexplicably played the ball into his own area.
He very nearly got in once again to complete a 30-minute hat-trick only for Boruc to crowd him out, but he was not to be denied his treble as the game became stretched in the second half.
With the score at 3-2, he completed his hat-trick with a simple side-foot volley to kill off Bournemouth's hopes of completing an unlikely comeback, while his 17th goal of the campaign followed just moments later when ran on to a Barkley backheel, shrugged off his marker and found the corner.
A heavy defeat, but Cherries show fight
Given their recent form, Bournemouth might have been expected to roll over after conceding after just 30 seconds, yet despite shipping six goals they will surely take some positives from this match.
Their three-man defence of Francis, Steve Cook and Tyron Mings was ripped to shreds in the first half, but the switch to a four-man defence after the break paid instant dividends.
With former Liverpool youngsters Jordon Ibe and Brad Smith introduced for the second period, Bournemouth looked sharper from the outset.
They created a host of openings before King's brace - the first a nice chest down and finish from Jack Wilshere's lofted pass, the second a poacher's strike from Ryan Fraser's centre - set up a grandstand finish that was only derailed when Lukaku bagged his third.
"The start of the game killed us. You can't concede so quickly, it gives you a mountain to climb," said Howe, whose side have now lost five of their last six away league games.
"We got right back in the game, then conceded the fourth to knock the stuffing out of us."
Man of the match - Romelu Lukaku (Everton)
The easiest man of the match award to select this season. Lukaku was immense from start to finish, each of his goals demonstrating the mix of finesse, skill, brute strength and instinct that makes him such a handful for opposition defences. The assist for McCarthy's goal was the cherry on the cake in a near faultless display
'If we keep this up, everything is possible'
Everton striker Romelu Lukaku: "It's disappointing we conceded the goals, but we showed mental resilience.
"We need to see the clips again because if you want to improve we need to get rid of the little problems.
"We have to believe that we can catch the teams ahead of us. We need to keep on winning and working hard for each other."
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Koeman reflects on 'crazy game of football'
Everton manager Ronald Koeman: "That's not a normal result. It was a crazy game.
"Both teams changed systems - it was a tactical battle. They made it really difficult for us. We dropped in intensity after half-time, but the reaction to them getting back to 3-2 was good.
"We scored great goals. Romelu showed that he is one of the best finishers in football. Scoring is one of his big qualities. We need to support him, to create. We can improve, but he showed his best today. I'm not surprised because we see it in training.
"We are getting closer to the top six. We have momentum and we are confident. It's a long way to go until the end of the season. If we keep this up, everything is possible."
Bournemouth manager Eddie Howe: "We tried to do something different today, but it didn't work. Three injuries to defenders have made it difficult.
"We have been looking over our shoulder all season. We want to get to 40 points as quickly as possible."