Feilhaber scored with a sliding shot in the 45th minute after taking a centering pass from Frederik Krabbe.
Feilhaber, who credited his strong play to the confidence and fitness he gained playing for the USA this summer, had assisted on the winning goal in AGF's 1-0 win over Aalborg on opening day.
"It's a great time to be an Aarhus fan," said Feilhaber.
His performance at Randers earned him Man of the Match honors on Danish television with a record 94 percent of the fan votes.
American teammate Jeremiah White was in the United States and did not play against Randers.
Other Americans in the top level of Danish soccer:
-- Danny Califf (played the first 70 minutes for FC Midtjylland in a 4-0 loss at Silkeborg);
-- Marcus Tracy (Aalborg) and Michael Parkhurst (FC Nordsjælland) were off as the game involving their respective teams was postponed.