It seems clear that the violation of a copyright is not precisely the same thing as stealing, since it does not involve depriving another of his or her property. That does not automatically make it acceptable, however.
In principle, the violation of a just civil law (e.g. copyright) is wrong. However, many modern laws are purely penal, meaning that their violation does not involve a direct moral fault, but simply subjects one to the possibility of judicial punishment. It is not always clear whether a given law is purely penal or not.
Since copyright is simply an grant to a private entity of sovereign authority’s right to regulate commercial transactions, one could argue that royalties and the like are equivalent to indirect taxes, which are purely penal according to the most common view of modern theologians.
If one accepted this analogy, then while it would clearly follow that any behavior which caused notable harm to a copyright holder (e.g. making a copyrighted work publicly available without paying royalties) would be a form of unjust damage, it would seem that simple personal use of pirated software would not be morally illicit.
However, the argument is tentative, and I’m not entirely sure of it myself.
Another factor that complicates your situation is that you already have these games downloaded, so it is simply a question of actively playing them. Possession and use of pirated materials is not against the law, but the process whereby a computer activates a program may legally constitute “copying”, depending on certain technical considerations.
Reference for the premsies I’ve drawn from moral theology: