As Edna said, the origin of the expression "Nossa!" comes from the expression "Minha Nossa Senhora". Brazil is known as a catholic country and the expression "Minha Nossa Senhora" literally means "My [dear] Our Lady" (referring to the mother of Jesus).
So, in this context "Minha Nossa Senhora" and "Oh my God" have similar meanings, because can be used in the same situation. Considering this, in my opinion the short versions "Nossa!" and "OMG" are similar, although not exactly the same.
p.s.:
- Usually "Nossa" is translated in informal texts as "Wow!".
- The literal translation for "Oh my God" is "Ó meu Deus!" or "Meu Deus!", and those are popular expressions too.