In the first half, the Bees twice led through Bryan Mbeumo and Yoan Wissa, but Alexander Isak and Harvey Barnes hauled United level each time they fell behind.
But Nathan Collins' second half strike gave Thomas Frank's side the lead once more before Kevin Schade's late goal made sure of the three points for the hosts, as Eddie Howe's men slipped to defeat at the Gtech Community Stadium.
The game managed to evade the worst effects of Storm Darragh, which wreaked havoc on the English football schedule on Saturday, and United began well. Barely two minutes in, Sean Longstaff - in for Sandro Tonali in one of two changes made by head coach Howe following Wednesday's 3-3 draw with Liverpool - rattled the crossbar with a well-struck 20-yard drive after good work from Barnes and Isak in the build-up.
But seven minutes in, the hosts seized the advantage. Mbeumo collected a lofted pass on the right and cut inside before driving into the box and doing well to hold on to the ball. The Magpies' backline couldn't prevent him from getting his shot away, though, and it was perfectly placed, arrowing past a stationary Nick Pope and finding the top right-hand corner.
The Bees' lead didn't last long, though. After Isak had fired over, the Swede made sure when presented with an opening moments later. It was well-worked, with Longstaff feeding Isak, who in turn fed Bruno Guimarães, who shifted the ball to Jacob Murphy wide on the right. The winger cut inside and his left-footed cross-shot was heading wide until it was met by the diving Isak, who diverted the ball home with his head to restore parity.
Isak perhaps should have put the visitors in front soon after, when he pounced on a defensive error and rounded goalkeeper Mark Flekken. Before he could apply the finishing touch, the hosts' stopper recovered to claim the ball, shortly before Guimarães' drive flew wide in an open, absorbing first period.
Before the half hour, United fell behind again. Barnes - the other player brought into the line-up by Howe - misplaced his crossfield pass and allowed Wissa to drive towards the box. He cut inside slightly and his curling effort just evaded Pope's grasp on its way into the same corner his teammate Mbeumo had found earlier.
It took Barnes just a few minutes to atone for his error, though, as he produced his side's second leveller of the afternoon. Murphy's cutback from the right found the former Leicester City wideman, who adjusted his feet and fired low across Flekken to make it 2-2.
Murphy, with two assists to his name, then headed a Barnes cross over as the game showed no sign of slowing down, even after the restart. Pope managed to turn a Wissa attempt, which had taken a deflection, onto the post barely three minutes into the second half, before Barnes tried unsuccessfully to bend one into the corner after drifting in from the left.
But before the hour, Brentford moved ahead for the third time. It came following a long punt forward from Flekken which wasn't dealt with, allowing centre back Collins to roll home beyond Pope from just outside the six-yard box.
Howe introduced Tonali and Anthony Gordon from the bench almost immediately, replacing Joelinton and Murphy, and Gordon whipped a shot over the bar following a corner as United looked to carve out a chance for a third equaliser.
With 17 minutes remaining, Callum Wilson and Miguel Almirón were sent on to help with the hunt for a leveller, but a resolute Bees backline made it tough for the Magpies to create goalscoring opportunities. They did have one big chance in the closing minutes though, and it fell to substitute Tonali. The Italian midfielder met Isak's cross in the box, but his header cleared the crossbar.
And a bitterly disappointing day for United was compounded in the final moments as substitute Schade was played in by Mbeumo to fire home a fourth for the Bees, who move into sixth place in the Premier League table - three points above Newcastle.
