Steve Clarke's side secured their fourth win of their pre-season campaign and will next play Atromitos at The Hawthorns on July 29
The Premier League side began well at Pride Park, with summer acquisition Nicolas Anelka going close after just four minutes.
Despite their dominance, Steve Clarke's side were thwarted in their attempts to take the lead and would have to wait until six minutes before the break to open the scoring as Dorrans drove a long-range effort past Derby goalkeeper Lee Grant.
The second half saw West Brom create several good chances, with Anelka again bringing a good save from Grant before James Morrison also tested the 30-year-old.
The home side almost grabbed a late equaliser through Michael Jacobs three minutes from time, forcing a good save from Ben Foster, but the result was sealed in added time when West Brom added a second.
Sweden international Rosenberg was on hand on turn home a Billy Jones cross and secure the club's fourth win of pre-season.