The hosts took a two goal lead in the first half, and thanks to some resilient defending they prevented a late fight back to solidify their position in second in the Premier League.
Liverpool looked the brighter of the two sides in the opening minutes, playing much of their football in the United half.
It was the hosts who opened the scoring however, with Rashford capitalising on some slack defending in the 14th minute.
David De Gea played the ball up field directly to the head of Romelu Lukaku, who beat Dejan Lovren in the air to slide in Rashford who got the wrong side of Alexander-Arnold. Entering the area, he cut back inside with the calmest of chops, firing across Loris Karius and into the bottom right-hand corner.
With the visitors desperate to get back in the game immediately, Roberto Firmino forced a corner with a shot from range. The ball was delivered to the near post, and despite Virgil Van Dijk finding himself free six yards from goal, he couldn't make a clean connection and fired wide.
Only moments later the Red Devils were back down the other end of the pitch, with Lukaku winning an almost identical header against Lovren to put Alexis Sanchez in on goal.
Van Dijk and Lovren haphazardly combined to clear the danger only as far as Rashford, who saw his first-time shot deflected viciously away from Karius to double the score.
The script looked written for a typical backs against the wall performance from a Jose Mourinho side, but they continued to look more efficient going forward as Juan Mata found himself with acres of space in the box only minutes later.
His attempted overhead kick was wide of the mark, leaving the visitors counting their blessings.
Liverpool came out for the second-half with a mountain to climb but it wasn't a challenge they were going to turn away from.
With United sitting deep, Firmino was first to try and get the ball in behind when he dinked the ball over the defence into the path of Mo Salah, who had been contained superbly in the first half.
Salah was first to the ball in the area, with Ashley Young playing a dangerous game as he leant all over the Liverpool forward who stayed on his feet. It came to no avail though, as he lost possession.
Next to call penalty was Andy Robertson, who got around the back of the United defence to fire a driven ball across the area. His cross struck the arm of Antonio Valencia, but it's hard to argue with the referees decision that his body position was natural.
The Reds were handed a lifeline 25 minutes from time though, with Eric Bailly connecting badly with the ball to clear a simple Sadio Mane cross, seeing the ball fall into his own net.
Liverpool had to be patient with the clock ticking on, and the Reds had arguably their biggest penalty shout of the match when Mane was brought down by substitute Marouane Fellaini, but nothing was given.
Mane was again the man in action when he slipped between the defence in added time, but Bailly recovered well and forced only a corner.
The final chance of the game fell to the man Liverpool fans would've hoped for, but with the ball falling high out of the air, Salah couldn't quite adjust to get over the ball, and he blazed over the bar.