The Armenian, who cost €27.5 million [£23.4m] from Shakhtar Donetsk this summer, came off the bench against Eintracht Braunschweig as the Schwarzgelben secured a second win out of two games in the 2013-14 Bundesliga campaign.
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"The crowd's [reception] was perfect," Mkhitaryan told Ruhr Nachrichten. "The atmosphere here is great and I do not lack support here. I'm enjoying every day.
"I am very happy, because we still won in the end. The most important thing is the three points. I was very impressed and excited before the game as it was my debut.
"I can certainly play better and I should have grabbed a goal. I hope that I'll be in better shape quickly. The injury is certainly in the past - I am fit."
Dortmund's next game is a Bundesliga home tie with Werder Bremen on Friday, with kick-off at 19:30BST.