The Gunners came into Sunday's encounter on the back of a disappointing home Champions League defeat to Olympiacos and with the spotlight firmly back on manager Arsene Wenger.
However, Arsenal tore United apart with two early goals in the space of 74 seconds, a backheel from Sanchez and a cool finish by German playmaker Mesut Ozil, with Sanchez cracking home a superb third to put the game to bed after only 20 minutes.
United improved in the second half, but it was too little too late.
The impressive victory moves Arsenal above United on goal difference, and they are now only two points behind new leaders Manchester City heading into the international break.
Wenger had called for a response from his side - and that is exactly what they delivered with a blistering start.
Arsenal were on the offensive from kick off and swept ahead after just five minutes.
Aaron Ramsey fed Ozil down the right channel, the German World Cup winner ghosting through the penalty area before cutting the ball back to the near post.
Sanchez arrived on queue ahead of the defenders to backheel a sublime finish past David de Gea.
Before United could regroup, they were immediately opened up again.
This time it was down the left, with Walcott dashing past Daley Blind
The England forward played a ball back out to Ozil, who was waiting patiently on the penalty spot to dispatch a cool sidefoot finish into the bottom left corner.
Following the electric start, the tempo of the match dropped, as United look to build a spell of possession.
It was, though, only a brief respite before Sanchez delivered a world-class finish to put Arsenal 3-0 ahead.
Another quick break found Walcott, who in turn fed the ball out to the left.
Sanchez brushed off the challenge of Matteo Darmian before sidestepping Chris Smalling and firing the ball into the top-right corner.
It was now a case of damage limitation for Louis van Gaal's men, who had come into the match on the back of three straight league wins.
Arsenal should have moved further ahead on 34 minutes when Sanchez chipped a ball over to Aaron Ramsey unmarked at the far post, but the Wales midfielder could only blast his shot high into the Clock End.
As half-time approached, Petr Cech produced a stunning block to deny Anthony Martial at point-blank range after the in-former French forward turned smartly in the penalty area.
Arsenal continued where they had left off at the start of the second half, with Ozil flashing a low drive just wide following a free-kick.
At the other end, Cech got down well to turn a low, curling shot from Wayne Rooney behind before then coming out to make a brave block at the feet of Bastian Schweinsteiger.
During the closing stages, Sanchez was forced off with what looked like a groin problem.
In stoppage-time, Arsenal substitute Alex Oxlade-Chamberlain was unfortunate not to make it 4-0 when his angled chip floated over the keeper, but hit the crossbar.
TWEET OF THE MATCH
Piers Morgan @piersmorgan - 'Best first half I have ever watched as an Arsenal fan
Played United off the pitch & half way back down the M1.'
https://twitter.com/piersmorgan/status/650700719989911552
PLAYER RATINGS
ARSENAL
Petr Cech: 8 (out of 10)
Hector Bellerin: 7
Per Mertesacker: 7
Gabriel: 7
Nacho Monreal: 7
Santi Cazorla: 7
Francis Coquelin: 7
Aaron Ramsey: 7
Mesut Ozil: 8
Alexis Sanchez: 9
Theo Walcott: 8
SUBS
Olivier Giroud (for Walcott, 75): 6
Alex Oxlade-Chamberlain (for Ozil, 75): 7
Kieran Gibbs (for Sanchez, 81): 6
MANCHESTER UNITED
David De Gea: 5
Matteo Darmian: 3
Chris Smalling: 5
Daley Blind: 4
Ashley Young: 4
Michael Carrick: 5
Bastian Schweinsteiger: 5
Juan Mata: 6
Wayne Rooney: 5
Memphis Depay: 5
Anthony Martial: 6
SUBS
Antonio Valencia (for Darmian, 45): 5
Marouane Fellaini (for Depay, 45): 6
James Wilson (for Mata, 82): 5
STAR PLAYER
Alexis Sanchez: Left the field to a deserved standing ovation after a wonderful individual display
The Chile international followed up his early flick with an exceptional strike, while his movement and skill proved a nuisance right up until his substitution.
MOMENT OF THE MATCH
Arsenal's start and, most impressively, the way Sanchez found the net
Quick, sharp build-up play saw Mesut Ozil cutback from the byline to the six-yard box
Unmoved by the attention of United's defence and David de Gea, Sanchez's quality was there for all to see with his instinctive flick home.
VIEW FROM THE BENCH
Matteo Darmian, Chris Smalling, Daley Blind and Ashley Young is not a back four that fills you with confidence
Injuries have taken their toll at United and that defensive line-up struggled in the first half, just like Bastian Schweinsteiger and Michael Carrick ahead of them
The directness, pace and pressing of Arsene Wenger's side made them pay.
MOAN OF THE MATCH
Billed as a big game but seven minutes in it was over as a contest
Louis van Gaal's side were ruthlessly punished for their early sloppiness in what was a first-half cake walk for the Gunners.
WHO'S UP NEXT
Watford v Arsenal (Premier League, October 17)
Everton v Manchester United (Premier League, October 17)
Source : PA
Source: PA