Yes a 7770 is going to be fine. Just have realistic expectations of it, don't assume it will max everything out because, well, it won't.
Considering my old 6670 can play games just fine your 7770 will also. (albeit the 6670 is starting to show its age, plays Civ V and Borderlands 2 on medium/high ish settings just fine.) (Though neither of those titles are newish)