Arsenal gave themselves some hope when Pierre-Emerick Aubameyang pulled one back before half time, but they could not find a way past Sergio Romero for a leveller and substitute Anthony Martial killed things off for good late on.
It took until just before the half-hour mark for the game's first shot to arrive, with Alex Iwobi succeeding only in picking out Romero at his front post following some nice build-up play between Aaron Ramsey and Alexandre Lacazette.
The Gunners had been the side on top up until that point, but United soon sparked into life and they scored twice in quick succession to completely change the mood around the Emirates Stadium.
Jeered every time that he touched the ball in the first 31 minutes, Sanchez got his revenge by rounding Petr Cech and converting into an empty net from a tight angle to open the scoring.
Lukaku claimed the assist after playing a disguised through-ball for Sanchez, and the Belgian also played a direct part in the Red Devils' second a couple of minutes later.
United hit their opponents on the counter and Lukaku had acres of space to run into down the right, before looking up and picking out Lingard, who controlled the ball and then tucked it out of Cech's reach.
The Gunners had progressed in seven of their last eight fourth-round ties prior to tonight, though, and they were right back in this contest when carving United open for the first time shortly before half time.
Ramsey latched on to Iwobi's pass on the overlap, held off Nemanja Matic and played the ball across goal for Aubameyang to tap in at the back post.
Arsenal enjoyed a spell well on top at the end of the first half and maintained that pressure early in the second period, coming close to a leveller through Ramsey's header which Romero brilliantly tipped over the crossbar.
Unai Emery brought on Mesut Ozil for the final half-hour, but a long stoppage for Koscielny to receive treatment to a facial injury - sustained after being accidentally caught in the face by Lukaku's boot - meant that the tempo of the match dropped.
With United focusing more on counter-attacks as the match entered its final quarter, Solskjaer turned to Martial and Marcus Rashford from the bench to provide some pace in behind.
That tactic paid off as Martial was left with a simple finish eight minutes from time once Cech parried Paul Pogba's powerful drive into his path.
Arsenal were unable to find a way through on goal in the 10 minutes of added time, with the biggest flashpoint coming off the ball as Lingard appeared to be hit by a coin following a coming together with Sead Kolasinac near the corner flag.
ARSENAL (4-3-3): Cech; Maitland-Niles, Sokratis (Mustafi 21'), Koscielny (Guendouzi 64'), Kolasinac; Torreira, Ramsey, Xhaka; Aubameyang, Lacazette, Iwobi (Ozil 64')
MANCHESTER UNITED (4-3-3): Romero; Young, Lindelof, Bailly, Shaw; Herrera, Matic, Pogba; Lingard (Jones 88'), Lukaku (Rashford 72'), Sanchez (Martial 72')