Eric Bailly sent the ball into his own net 24 minutes from time to give Liverpool hope, but the in-form Merseysiders failed to register any further attempts as they fell to just a third defeat in 28 matches.
Rashford scored his first goal of the match from the first attempt of any note at either end, nodding Romelu Lukaku's flick-on into his path before cutting inside Trent Alexander-Arnold and firing home with his right foot.
Goal number five of the Premier League campaign for Rashford was soon followed by a sixth, and it was another composed finish from the youngster as he swept home the loose ball after Lukaku's through-ball was cut out.
The Englishman's second goal came moments after Virgil van Dijk had squandered a big chance at the other end of the field, failing to make proper contact with a corner when swooping in from five yards out.
That would prove to be the Reds' only real opportunity in the first half, managing just a couple of weak shots from range through Roberto Firmino.
United could well have been home and dry before the interval, as Juan Mata sent an overhead kick narrowly wide at the end of a flowing move when he had time to first bring the ball under control.
The visitors, on a three-match winning streak on their travels without conceding prior to today, failed to muster too much of a response in the opening 15 minutes of the second half.
Van Dijk found the roof of the net with his latest headed effort and James Milner sent a shot wide from distance as frustration started to creep in.
Ashley Young was then a little fortunate to avoid giving away a penalty after dragging back a quiet Mohamed Salah inside the box, and Antonio Valencia likewise when Andrew Robertson's cross struck his arm.
Following another quiet spell in the game, the whole mood around Old Trafford completely changed with a quarter of the match to go as Sadio Mane's cross was diverted into his own net by Bailly.
The centre-back, who passed a fitness test to start the game, got his intended clearance horribly wrong near the front post, using his calf to become just the seventh player to beat David de Gea here this season.
Liverpool saw plenty of the ball in the remainder of the match but could not create any further chances, with United successfully navigating the seven minutes of added time for another crucial home win.
MANCHESTER UNITED (4-2-3-1): De Gea; Valencia, Bailly, Smalling, Young; Matic, McTominay; Mata (Lingard 89'), Sanchez (Darmian 96'), Rashford (Fellaini 70'); Lukaku
LIVERPOOL (4-3-3): Karius; Alexander-Arnold (Wijnaldum 80'), Van Dijk, Lovren, Robertson (Solanke 84'); Oxlade-Chamberlain (Lallana 62'), Can, Milner; Mane, Salah, Firmino