Jeff Hendrick's volley sent Newcastle into the lead before Patrick Bamford's close-range header drew the hosts level before the half-time whistle.
Rodrigo's glancing header sent Leeds ahead and although Newcastle responded in quick fashion through Ciaran Clark's equaliser, goals in the closing stages from Stuart Dallas, Ezgjan Alisoki and Jack Harrison condemned United to a disappointing defeat.
Steve Bruce's side headed into their first Premier League encounter against Leeds since January 2004 searching for a third successive win after victories against Crystal Palace and West Bromwich Albion and faced a Leeds side without a win at Elland Road in five matches.
The Magpies, who scored inside 20 seconds in a 2-1 win against West Bromwich Albion, almost got off to another fast start as Isaac Hayden's intended cross from the left flank grazed the crossbar within the opening minute.
Karl Darlow was called into action in the opening stages as the goalkeeper pushed away Liam Cooper's goal bound header to safety following a teasing corner from Kalvin Phillips before Raphinha's drilled rebound was blocked by Hendrick.
In the 26th minute, the Magpies opened the scoring as Fraser's cross from the left wing was cleverly flicked towards the back post by Callum Wilson, with Hendrick on hand to place the ball home.
Raphinha went close to equalising for the Whites as the Brazilian's vicious first-time effort towards goal was importantly beaten away by Darlow.
The Magpies were dealt a difficult blow in the 35th minute as Leeds levelled through Bamford. After Rodrigo's looping header smashed off the crossbar, the ball fell kindly into Bamford's path who reacted quickly to nod the ball into an empty net.
In the second half, an early penalty appeal was waved away by referee Simon Hooper as Wilson fell inside the penalty area after a challenge with Cooper but after a VAR check, a spot-kick was not awarded.
Newcastle fell behind in the 61st minute as Harrison's cross from the byline found Rodrigo inside the penalty area before the Spaniard directed his header into the far right-hand corner.
The visitors went narrowly close to equalising minutes later as Wilson almost completed a neat counter-attack. After a quick move consisting of Fraser and Jacob Murphy, the latter sent a teasing pass into Wilson's path but the striker's first-time effort was tipped over the crossbar by Meslier.
However, United managed to draw level in the 66th minute as Clark rose well from Ryan Fraser's corner, directing a powerful header into the left-hand corner to record his first goal of the season.
In the 77th minute, though Leeds regained their advantage with another header, this time by Dallas. After Newcastle lost possession in a dangerous position, Matesuz Klich's cross was floated towards the back post and the Northern Ireland international was left unmarked to nod the ball beyond Darlow.
Newcastle pushed forward in hope of an equaliser but were punished in the 85th minute as Alioski doubled the hosts' lead following a clinical counter-attack as the Macedonian was teed up by substitute Pablo Hernández before firing the ball into the roof of the net.
Harrison then added a fifth in the closing stages as the 24-year-old, picking the ball up inside his own half, drove towards the United defence before rifling a ferocious effort into the top right-hand corner.
Newcastle United: Karl Darlow, Jacob Murphy, Isaac Hayden, Federico Fernández, Ciaran Clark, Jamal Lewis, Jeff Hendrick (Emil Krafth 62), Sean Longstaff, Ryan Fraser (Miguel Almirón 75), Joelinton (Dwight Gayle 73), Callum Wilson.
Subs not used: Mark Gillespie, Jonjo Shelvey, Matt Ritchie, Andy Carroll.
