As part of the deal, the Red Devils allowed Henrikh Mkhitaryan to move in the opposite direction.
"I'm very happy that we could finish this deal and I'm very happy to be here", said Mkhitaryan.
Arsenal allowed Sanchez to join Manchester United on Monday to avoid losing the Chile global on a free transfer when his contract expired at the end of the season.
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.
United are second in the domestic standings, 12 points behind Manchester City, and will hope Sanchez can help them cut the gap on their neighbours whilst also helping them progress in the Champions League, where they face Sevilla in the last 16.More news: Nicolas Otamendi sings new contract
Wenger rejected the theory that Sanchez was disrupting the harmony in the Arsenal team in his final months there.
He wrote: "There are people (former club players) who have spoken with no knowledge of what happens inside the club and cause damage".
'I've not had the chance to get to know them personally or spend time with any of my teammates up to now, but of course I know them from having played against them in the past, ' said Sanchez.
Alexis Sanchez is one of the best attacking players in the Premier League if not the world.More news: Rating Roundup: Inphi Corp (IPHI)
"Now that I'm here, I'll do my best for this club to create history", he said.
Mkhitaryan, 29, has played 63 times for United after arriving from Borussia Dortmund in 2016, scoring 13 goals, although he has recently struggled for game time under Jose Mourinho.
Manchester United fans across the world would wish to see their new signing chasing the ball in United iconic number seven that was previously worn by United legends namely, Erick Cantona, David Beckham and Christiano Ronaldo who is also reported to be considering a move back to United.
Sanchez in July 2014 joined Arsenal from Barcelona for about ￡30 million (US$41.9 million), choosing a move to London amid interest from Liverpool.
"He will bring his ambition, drive and personality - qualities that make a Manchester United player".More news: Adele joins anti-Trump women's march