Armenia international winger Mkhitaryan, 29, will head to the Emirates after he fell out of favour with Manchester United manager Jose Mourinho during 18 months at Old Trafford.
Mkhitaryan joined United for around £30 million in 2016 after three years at Borussia Dortmund.
The highlight of his stay in Manchester was scoring in the Europa League final victory over Ajax last season and also helped United win the League Cup.
“I would like to wish Henrikh all the success and happiness that I am sure he is going to get. He is a player that we will not forget, especially for his contribution to our Europa League victory,” added Mourinho.
Mkhitaryan told the Arsenal website he had “always dreamed of playing for Arsenal”.
“Now that I’m here, I’ll do my best for this club to create history,” he said.
Arsenal boss Wenger said he was happy to acquire Mkhitaryan in the deal.
“Henrikh is 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.”
The Sanchez-Mkhitaryan deal is the latest move of a January transfer window during which Liverpool sold Philippe Coutinho to Barcelona for 160 million euros and signed central defender Virgil Van Dijk from Southampton for £75 million (85.5 million euros).
Reports in Germany say striker Pierre-Emerick Aubameyang could be on his way to Arsenal from Borussia Dortmund.