The 24-year-old, who was linked with a move to Liverpool before deciding to join Jurgen Klopp's side, comes in after Bayern Munich activated Gotze's release clause.
However, the Armenian international insists that he is not a like-for-like replacement for the 21-year-old.
"[Gotze] is a very good player, but he's a different type of player to me," he told reporters. "Mkhitaryan is Mkhitaryan, and Dortmund is Dortmund. This atmosphere, these fans - it is unique in the whole of Europe.
"I've joined a very good team and the club have taken very good care of me, which makes it very easy and is very important for a new player.
"I hope that Dortmund never regret this transfer for a second, and that they don't think any cent is wasted."
Mkhitaryan arrived at the Westfalenstadion from Shakhtar Donetsk in a deal worth €24m (£20.5m).