Eddie Howe's side netted three times in the first-half, with Harvey Barnes' header breaking the deadlock before quick-fire strikes from Jacob Murphy and Alexander Isak extended the Magpies' advantage further, who escaped an early scare as Ivan Toney's goal was disallowed following a VAR review as Bryan Mbeumo strayed into an offside position.
Vitaly Janelt's strike shortly after the restart and Yoanne Wissa's bending effort looked to set up a nervy finale before Bruno Guimarães' close-range rebound, shortly after Newcastle saw a penalty overturned, ensured a seventh-placed finish for United, who will secure qualification for the 2024/25 UEFA Conference League should Manchester United lose against Manchester City in Saturday's Emirates FA Cup final.
United looked to have conceded inside the opening two minutes as Toney, who made his Premier League debut for the Magpies in 2015, slotted home from Mbeumo's low cross into the penalty area but, after a VAR check, the goal was disallowed after the latter had initially strayed into an offside position.
Murphy went close to opening the scoring on his 150th Premier League appearance as the versatile winger fired a dipping shot from the edge of the penalty area wide of the right-hand post.
Pope, making his first Premier League outing since December, produced an important save to deny Toney minutes later after the Bees striker latched onto Mbuemo's pass inside the box before his resulting low effort was blocked by the right foot of the Magpies goalkeeper.
Joelinton, making his first league start since New Years' Day, missed a glorious opportunity to fire United in front as the Brazilian was picked out by Isak at the back post following the Swede's impressive run along the right flank but the South American midfielder's low shot was directed wide of the left-hand post.
In the 22nd minute, Barnes fired Newcastle ahead with his fifth Premier League goal of the 2023/24 campaign as the Burnley-born winger rose high from Guimarães' excellent cross into the box before guiding his header into the top left-hand corner.
Sean Longstaff fired the ball into the net minutes later, slotting a low effort beyond Bees goalkeeper Mark Flekken following Isak's pass but the North Shields-born midfielder's goal was ruled out for offside as Isak was adjudged to venture into an offside position.
In the 36th minute, Murphy doubled the Magpies' lead, the club's 100th goal across all competitions this season. Despite Isak being stopped by Ethan Pinnock in the build-up, the Swedish frontman recovered the loose ball before his low cross found Murphy, who tucked the ball home at the back post.
Two minutes later, Newcastle added a third as Isak turned from provider to goalscorer. As Murphy regained possession on the right flank, his pass found Guimarães who then shuttled the ball into Isak's path, who placed a first-time effort into the far left-hand corner, his 25th goal of the season across all competitions.
Newcastle's hopes of a clean sheet were distinguished three minutes after the restart as Brentford pulled a goal back through Janelt's strike. After Mathias Jensen's pass found Wissa inside the box, the latter squared the ball into the German's path, who fired the ball into the bottom left-hand corner.
Isak went close to restoring United's three-goal cushion after the hour mark after Fabian Schär's free-kick struck the wall before diverting into his path but the 24-year-old was unable to grab his 12th league goal in as many games after dragging his effort wide of the target.
Pope made an excellent save to prevent Toney from scoring, getting down to his right before crucially palming away the striker's towering header.
In the 70th minute, however, Wissa halved the deficit in impressive fashion as the Congolese forward collected a pass from Toney and, after stepping inside, struck a powerful beyond a stretching Pope.
Newcastle were awarded a penalty in the closing stages after Lewis Hall was tripped by Mbeumo following an impressive run by the teenage left-back but VAR decided the clip came outside the box to the frustration of the United squad.
However, in the 77th minute, Guimarães scored from the following passage of play as the Brazil international reacted quickly after Isak's free-kick was parried by Flekken before tucking the ball into the net, racing towards the elated away supporters before placing a Newcastle-branded bucket hat onto his head.
Following Chelsea's 2-1 win against AFC Bournemouth, Newcastle missed out on a top-six finish by three points but could return to Europe next season should Manchester United fail to win the Emirates FA Cup next weekend, competing in the UEFA Conference League.
