Kluivert's early goal rewarded Andoni Iraola's men for a lively start on Tyneside, but Bruno Guimarães' header swiftly brought the hosts level.
Bournemouth - who had been on a similarly impressive run of form going into the contest - proved tricky opposition, and took the lead again just before half time when Kluivert rolled home his side's second.
Two stoppage time goals then added further gloss to the scoreline, with Kluivert's 20-yarder sealing his treble before Milos Kerkez slammed home a fourth to wrap it up.
This was far from a fluent display from a Magpies side that had been in such irrepressible form over the last month, and the Cherries made them pay in a tetchy game at St. James' Park to extend their own unbeaten run to ten Premier League games.
The visitors, depleted by injury in recent weeks, began brightly and perhaps should have moved in front even before their early opener. Barely four minutes in, David Brooks' inswinging cross was met by Dango Outtara, who forced a fine save from Martin Dúbravka before Antoine Semenyo blazed the rebound over.
It was a glorious chance but they only had to wait a further minute or so to take the lead. Semenyo got behind the Magpies' backline and his cutback from the left found Kluivert - the son of former Newcastle striker Patrick - confidently swept home the opener.
Joelinton whipped a shot well over as United sought a response and 25 minutes in, they got one. Lewis Hall's corner found Guimarães unmarked and his header crept into the bottom corner despite the best efforts of Kepa Arrizabalaga, who got a hand to it but couldn't scramble it away.
Kluivert's inswinging cross almost sneaked in at the far post and Ryan Christie miscued when well placed, before Outtara nodded a Lewis Cook centre over with the Cherries retaining their threat going forwards, but United were dangerous when they attacked, or counter-attacked, with typical pace. Anthony Gordon saw his strike deflected past the upright in one such break.
But Kluivert's second of the day just before half time left Howe's side with work to do. Christie robbed Guimarães in the Magpies' half and Outtara's through ball for the Dutchman - who found the bottom left-hand corner of the net - was just as neat as the eventual finish.
Howe introduced Fabian Schär in place of Sven Botman at the interval, but the Swiss centre half's first contribution was to inadvertently presented Bournemouth with a chance. His wayward crossfield ball was picked up by Christie, whose curling effort was comfortably held by Dúbravka. The Scottish midfielder tried again soon after having eased Hall off the ball 25 yards out, but his effort was blocked.
On the hour, the Cherries struck again, and only a slice of fortune prevented Newcastle from falling further behind. Outtara's run down the right had culminated in a close call for United as his shunted effort came back off the upright, and despite hacking the ball away, seconds later Outtara had the ball in the net after turning in Brooks' fine cross from the right. The subsequent VAR call, though, gave the Magpies a late reprieve; the ball was adjudged to have gone out before Sandro Tonali's clearance just prior to the goal, meaning the visitors' third was chalked off.
That call brought roars of encouragement from the Newcastle faithful and Howe responded by sending on Kieran Trippier and Joe Willock, and his side responded with a spell of pressure without really testing Kepa between the posts. At the other end, Dúbravka clawed one off the line before Gordon's sensational volley was blocked just as it looked set to ripple the net.
Will Osula was sent on just before the seven minutes of stoppage time in the hope of snatching a point, but almost as soon as he entered the fray, Bournemouth wrapped up the win. It was a terrific strike from Kluivert, who collected Tyler Adams' lay-off after Gordon's wayward pass and fired home to complete his treble.
And Kerkez's arrowed finish across the helpless Dúbravka then made it four in the dying moments, compounding a thoroughly miserable afternoon for United.
