Udogie swept home Son's cross to put Spurs in front before the South Korean forward supplied a similar assist for Richarlison to double the hosts' advantage before the break.
The Brazilian's second of the day on 60 minutes gave Ange Postecoglou's side an unassailable lead, and though Eddie Howe was able to bring fit-again duo Callum Wilson and Sean Longstaff off the bench on their returns to action, this was another away day to forget for the Magpies, who conceded a fourth when Son rolled home from the spot with six minutes left.
Wilson set up Joelinton to fire home a consolation in added time for United, but that was as good as it got for Howe's charges in the capital.
Postecoglou's men, whose formed had trailed off since their blistering start to the season, probed for much of the first half. Bruno Guimarães' vital block stopped countryman Richarlison from shooting early on and the Spurs forward then fired just over, with Fabian Schär's slight touch perhaps diverting the effort off target.
Guimarães' raking drive from Joelinton's cutback whistled just over and in a fairly open start, Newcastle carved out an even better chance. Anthony Gordon was released on halfway and made his way to the box unchallenged but his pass across the face of goal for Alexander Isak was intercepted by Ben Davies, who get just enough on the ball to stop the Swedish forward from tapping into an open goal.
It was a fine intervention and it led to a spell of Spurs pressure. After Cristian Romero's goalbound header was hacked off the line by Miguel Almirón, the hosts took the lead. It was made by Son, whose teasing ball across goal cut out Martin Dúbravka, and full back Udogie arrived right on cue to slam home from close range.
There were still openings for Newcastle - Joelinton had a shot charged down just after the first goal - but they weren't as clear as the opportunities that Spurs created. Pape Matar Sarr scuffed Brennan Johnson's cross wide but, after Almirón had prodded Gordon's lay-off straight at Guglielmo Vicario in the home goal at the other end, the hosts had their second. Again, it was Son who made it, beating Trippier in the penalty area and cutting the ball back from the byline for Richarlison to direct into the bottom right-hand corner.
It could have got even worse for United before the break, with Johnson's fierce strike skimming the post and both Dejan Kulusevski and Richarlison going close. The latter's chance was the better one, as he stooped to head the former's deflected cross into the net, but somehow failed to make contact.
There was a response from the Magpies at the start of the second half, with some sustained pressure and set pieces in the final third. Gordon hooked a Trippier cross over the bar, and within seconds Davies had to put himself in the way of another centre from the England full back before Gordon pounced on it.
But while Spurs had lost some of the fluency they had shown before the restart, Newcastle struggled to find a way to make their improvement count, and just before the hour Postecoglou's men put the contest to bed with their first real attack of the half. Pedro Porro's lofted pass over the top left Jamaal Lascelles trailing and Richarlison collected it, squeezing his left-footed finish under Dúbravka.
Johnson then thumped the upright once more, hammering the woodwork with a cross-shot, and Son sent a back-post volley wide of the target as the hosts went in search of more. After Howe introduced Wilson and Longstaff - with the pair having recovered after a month out injured - Kulusevski and Son both tested Dúbravka, who later had to keep out a low drive from Porro.
But in the final ten minutes the Slovakian goalkeeper was penalised for bringing down Son as he looked to round him when played through on goal. Dúbravka went the right way, but the skipper's spot-kick was perfectly placed - though the United stopper did well to prevent Oliver Skipp from making it five moments later.
Wilson's lay-off was coolly despatched by Joelinton in the game final moments, but it was far too little, too late for Newcastle, whose focus now turns to Wednesday's all-important Champions League clash with AC Milan at St. James' Park.
