Since the start of the January transfer window, Sanchez has been linked with an exit from the Emirates Stadium, but it soon emerged that a transfer to Old Trafford - rather than Manchester City - was in the offing for the Chilean international.
Reports indicated that Arsenal boss Arsene Wenger was keen on replacing Sanchez with Mkhitaryan, but the Armenian international's representative Mino Raiola suggested that a move was far from guaranteed.
However, after both players were granted new work permits on Monday afternoon, United and the Gunners are now in a position to confirm that the pair have traded clubs.
United have not revealed the length of Sanchez's contract, but reports have suggested that the 29-year-old has penned a four-year deal with the North-West outfit.
Sanchez could make his debut for United when they face Yeovil Town in the FA Cup fourth round on Friday night.