United were not at their best but quickfire goals in the first half from Sanchez and Jesse Lingard, both of which were set up by Romelu Lukaku, gave them a 2-0 lead at the Emirates Stadium on Friday.
Arsenal hit back through Pierre-Emerick Aubameyang but could not find a way to beat David de Gea's deputy Sergio Romero again, with Anthony Martial's late third wrapping it up for United.
Victory for United means they have won eight games out of eight in all competitions under Solskjaer, while opposite number Unai Emery will rue injuries to defenders Sokratis Papastathopoulos and Laurent Koscielny.
Arsenal lost Sokratis to an early injury and things got worse for the Gunners when they conceded to an old hero with United's first effort in the 31st minute.
Sanchez scored in Arsenal's FA Cup final wins in 2015 and 2017 and he brilliantly rounded Petr Cech to finish from a tight angle after collecting a sublime reverse ball from Lukaku.
And the Belgium striker was involved again when United doubled their lead 149 seconds later.
Arsenal's offside trap failed for the second time in quick succession with a rapid counter-attack freeing Lukaku to square a pass that Lingard controlled before finishing low past Cech.
Emery's men gave themselves a lifeline before the break when Aubameyang tapped in at the back post following good combination play from Aaron Ramsey and Alexandre Lacazette.
Romero saved brilliantly from Ramsey after the restart before Koscielny was forced off with a facial injury following a long delay after being caught by Lukaku's boot.
And United booked their place in the fifth round when another counter-attack undid Arsenal in the 82nd minute. Paul Pogba's driving run ended with a shot parried by Cech and Martial tucked away the rebound.
Arsenal's soft centre exposed
There is no doubting Arsenal's attack is potent, but their defence was torn open twice in three minutes by a United side who rarely threatened before or after their double-salvo. Injuries to both starting centre-backs, Sokratis and Koscielny, will hardly help matters for Emery, who has already lost Hector Bellerin for the rest of the season. As for United, they keep on finding a way to win under Solskjaer even if they do not truly sparkle.
Lukaku reminds Solskjaer of his talents
Marcus Rashford's brilliant form in recent weeks has kept the fit-again Lukaku on United's bench, but he was back in the team and created two of his side's goals. Lukaku's playmaking talents are underrated but he showed he is more than a goalscorer with cute passes for team-mates Sanchez and Lingard. Solskjaer has a decision to make for United's next game.
Kolasinac positioning costs Arsenal
Sead Kolasinac has many qualities but concentration does not appear to be one of them. For both United goals in the first half, Arsenal's rampaging left-back was caught significantly out of position, his mistakes giving the Gunners a mountain to climb. He was also fortunate not to be sent off after pushing his head into the face of Rashford late in the game. Referee Craig Pawson opted to book both players, while amid a melee Ashley Young may also face a later punishment after he appeared to flick an object back into the crowd after it was thrown onto the field.
