While Sanchez joined Manchester United from Arsenal, Armenian international Henrikh Mkhitaryan went the other way in a rare transfer swap involving two leading English Premier League clubs.
"I am fulfilling a dream that I have since I was a kid: to play for the world's best team," Sanchez wrote in one of several Instagram posts.
Meanwhile Arsenal manager Arsene Wenger described the 29-year-old Mkhitaryan as "a very complete player."
"He creates chances, he defends well, he can absorb distances and he's very committed as well. I must say he's a player who has all the attributes," Wenger told the Arsenal website.
'The biggest mercenary in football'
The 29-year-old Sanchez had widely been expected to join rivals Manchester City in the summer transfer window but a deal fell through on deadline day.
Despite City reviving their interest in January, and Chelsea inquiring about a deal, the 29-year-old opted to join Jose Mourinho in the red half of Manchester.
As the saga played out former Arsenal defender Martin Keown called Sanchez "the biggest mercenary in football," while another former Gunner Ian Wright said in December the Chilean had "clocked off."
It has been widely reported that Sanchez's weekly salary at United will range from $420,000 to $700,000.
On Instagram, Sanchez did not reference anyone directly as he defended his record at Arsenal, writing: "There are people (former club players) 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 (Arsenal manager Arsene Wenger) to be in the team, because I wanted to be a contribution."
Mourinho believes Sanchez is "one of the best attacking players in the world."
"He will bring his ambition, drive and personality, qualities that make a Manchester United player and a player that makes the team stronger and the supporters proud of their club dimension and prestige," added the United manager.
FA Cup success
After three season at Barcelona, Sanchez joined Arsenal in 2014 for a reported fee of $44 million.
In his three and a half years in north London, the Chile international helped the club win two FA Cups and the Community Shield -- while also winning two consecutive Copa America titles for his country.
However, Arsenal's failure to finish in the Premier League top four last season and secure Champions League football meant Sanchez saw his future away from the club.
With Wenger unable to find a suitable replacement in the summer, the Frenchman refused to sanction a transfer for Sanchez.
That changed in January, when Mourinho offered Mkhitaryan as part of a swap deal.
"I believe we lose a world class player and gain a world class player," Wenger said in his Tuesday press conference.
"Overall I think Mkhitaryan has the qualities to integrate into our game. He has a good team attitude as well. I'm confident it will work.
"He can play in all kinds of positions. He is very versatile and it should help us going forward."
Mkhitaryan's route to Arsenal has taken him from his native Armenia, to Ukraine, Russia, Brazil -- where he went on trial to Sao Paulo as a 14-year-old -- then to Germany with Borussia Dortmund and now England.
Under now Liverpool manager Jurgen Klopp at Dortmund, Mkhitaryan won the German Supercup, scoring against Bayern Munich to help his club lift the title.
His habit of performing on the big occasion continued in his first season at Manchester United, scoring in the final of the Europa League as Mourinho's side beat Ajax 2-0.
Mkhitaryan also lifted the League Cup and Community Shield in his stint at Old Trafford.
"I would like to say a heartfelt thank you to Manchester United players and staff and especially the fans for your love, passion and support," the 29-year-old wrote on Instagram.
"I enjoyed having the honor of contributing to the history of your great club by winning the Europa League cup.
"I will always cherish the memory of that moment and that match. And of my Boxing Day scorpion kick and of hearing the song 'Whoa Mkhitaryan ...' every time I stepped onto the pitch."