Chelsea's slip-up at home to Southampton this evening means that United have closed the gap on the top four to six points with their fourth staight Premier League win since the sacking of Jose Mourinho.
The Red Devils had scored inside 10 minutes in two of their previous three matches and Rashford nearly gave them another quick start when he tried to find the bottom corner with a 30-yard free kick, Martin Dubravka reacting well after initially expecting a cross.
Newcastle began well themselves, with Christian Atsu a dangerous outlet on the left wing, and the Ghana international could have put the hosts ahead on a couple of occasions.
Atsu's fine piece of control set up a shooting opportunity from the left side of the penalty area on 13 minutes, but he shot tamely at David de Gea and his finishing lacked conviction again moments later after he had cut inside Victor Lindelof.
There was an even nervier moment in the United defence soon after as Phil Jones gifted the ball to Salomon Rondon on the edge of the box, only to recover and make a crucial last-ditch tackle before the in-form Newcastle striker could get a shot away.
The visitors were not finding it as easy to break down the Magpies as they did their previous opponents under Solskjaer, with the closest they came in the first half a deflected Anthony Martial shot which looped over Dubravka but onto the roof of the net.
While the Magpies held a firm defensive shape, they continued to present a threat on the counter and an overlapping run from DeAndre Yedlin yielded a cross that Rondon could not keep down with his head on 34 minutes.
Newcastle were made to regret their missed opportunities in the second period when United's depth of attacking firepower paid dividends.
Lukaku and Sanchez were called upon off the bench on 63 minutes, and within seconds the former had broken the deadlock as he pounced on Dubravka's spill of Rashford's free kick to poke home from inside the six-yard box.
Rafael Benitez's side have only scored four second-half goals in the Premier League this season and did not pose the same offensive problems to United after the break.
One of Benitez's own substitutes - Jonjo Shelvey - had two shots from the edge of the box, the first of which was well held by De Gea while the second was sliced wide.
The visitors put the game to bed 10 minutes from time when Lukaku and Sanchez combined to set up Rashford to slot past Dubravka from 10 yards, for his third goal in four starts as a centre-forward under Solskjaer.
Pogba missed a chance to continue his fine goalscoring form in stoppage time when he rounded Dubravka only to send his shot into the side-netting, but a third clean sheet of the season may have be more satisfying to United than a third goal.
Newcastle's eighth home defeat of the season leaves them only two points adrift of the relegation zone.
NEWCASTLE (5-4-1): Dubravka; Yedlin, Schar (Muto 81'), Lascelles, Dummett, Ritchie; Atsu, Hayden, Diame (Shelvey 53'), Perez (Kenedy 69'); Rondon
MAN UTD (4-2-3-1): De Gea; Valencia, Lindelof, Jones, Shaw; Herrera, Matic; Pogba, Mata (Sanchez 63'), Martial (Lukaku 63'); Rashford (Lingard 87')