The Chile international finalised a transfer to Old Trafford on Monday evening.
Last summer, the 29-year-old came close to joining United's neighbours Manchester City, but five months later, the attacker has opted to join Jose Mourinho's club, which he describes as the 'biggest in the world'.
In the last few months, Sanchez has faced criticism from former Arsenal players for a perceived lack of dedication on the pitch while playing for the Gunners, where he spent three-and-a-half years.
In a lengthy Instagram post, in which the attacker paid gratitude to the club, the Chilean added: "There are people who have spoken with no knowledge of what happens inside the club and cause damage.
"I must say I always gave 100%, until the last day, when I asked to the Mister (manager Arsene Wenger) to be in the team, because I wanted to be a contribution."
Sanchez could make his debut in United's FA Cup fourth-round tie against Yeovil Town on Friday.