United's winning run under the Norwegian is now eight games in all competitions and his decision to make five changes was vindicated as both Sanchez and the returning Romelu Lukaku had a huge impact.
Lukaku set up both Sanchez's opener and Jesse Lingard's second for the visitors in a two-minute spell just after the half hour mark.
Pierre-Emerick Aubameyang reduced Arsenal's arrears just before half-time and the Gunners had plenty of chances to force a replay, but were denied by Sergio Romero as the rested David de Gea was not missed.
Eight minutes from time, United sealed a place in the last 16 when substitute Anthony Martial tucked home the rebound after Petr Cech could only parry Paul Pogba's initial effort.
Sanchez was roundly booed on his first visit to the Emirates since moving to Old Trafford just over a year ago after running down his contract at Arsenal into its final six months.
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That move bagged the Chilean a highly lucrative deal, but Sanchez has failed to deliver on that investment with a loss of form and injuries contributing to a return of just four goals in 32 appearances before kick-off.
However, handed just a second start since Solskjaer took charge, Sanchez showed why he was once in such high demand as he latched onto Lukaku's pass before rounding Cech and finishing from a narrow angle for just his second goal of the season.
Solskjaer used Lingard in the middle of a front three with Marcus Rashford and Martial on either side as threats on the counter-attack to great effect in victory at Tottenham a fortnight ago.
A similar tactic with Lukaku and Sanchez this time in the wider areas worked a treat for United's second as Lukaku broke forward before his low cross was expertly dispatched by Lingard.
Arsenal were also exploiting the space in behind Lukaku at will, though, and finally made United pay two minutes before the break.
Aaron Ramsey burrowed into the penalty area and his cross was eventually turned home by Aubameyang after Alexandre Lacazette and Lucas Torreira failed to apply a telling touch.
Ramsey is also set to leave Arsenal after failing to reach an agreement over a new deal and the Welsh international again proved what a miss he will be as Romero made a brilliant save from point-blank range to deny him an equaliser early in the second period.
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Arsenal's momentum in the search for an equaliser was disrupted by a lengthy delay when captain Laurent Koscielny took an accidental blow to the jaw from Lukaku's boot and had to be replaced.
Having lost Sokratis Papastathopoulos to an early ankle injury, Unai Emery had to patch together a makeshift defence for the final half hour, but still opted to attack by introducing Mesut Ozil.
Arsenal enjoyed the vast majority of possession and territory, but couldn't get the equaliser as Romero again stood tall to parry Lacazette's powerful drive.
Solskjaer introduced Rashford and Martial for the final 18 minutes and the latter delivered the knockout blow to Arsenal by smashing home the third.
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