Sanchez was the focal point of Arsenal's attack but he had let his contract run out and was at risk of being let go on a free in the summer, before he and Henrikh Mkhitaryan switched places.
"He was very fond of me and he didn't want to let me go," Sanchez said to the Chilean media. "He told me: 'I want you to sign here because you are the best player in the team.'
"I wanted to continue growing as a player, more so as a Chilean, because it's difficult to get to Man United.
"No Chilean had played here. Arsene understood it at last, we ended up on good terms, he congratulated me and there was no problem,' Sanchez added.
Since joining Manchester United, Sanchez has struggled, scoring only once in six games while the team were dumped out of the Champions League by Sevilla.