Amini created the opening goal for Stephan Petersen and then doubled the visitors' lead after the break. Simon Kroon's late goal for the hosts was not enough to deny AGF the points.
Amini, who signed from Randers last month, was used in an attacking midfield position and was delighted after the win.
"To have a good start with the fans, it was a dream debut," he told AGF's YouTube channel.
"You know, I scored and to celebrate with the fans - we're at an away game and it felt like we were at a home game."