The Super Eagles would have gone out with defeat but came good in Volgograd.
Musa showed superb technique to fire Nigeria ahead on the half-volley before the Leicester player rounded keeper Hannes Thor Halldorsson to make it 2-0.
Iceland missed a late penalty awarded after a video assistant referee review, Gylfi Sigurdsson firing over.
Nigeria, who had started the day bottom of the group, are now second - three points behind leaders Croatia, who have already qualified for the knockout stage.
Iceland, who face Croatia next Tuesday, and Argentina, who play Nigeria at the same time, both have one point.