Hello bloodfallen.

If a game's information is not available in Steams API then there's no way to check for it.
There's other things to consider though: The sites rules require you to only enter giveaways for games that you don't own, yet. So you should have checked before entering. While I understand that hardly anyone does that and instead simply relies to the site's settings that are just meant to help with that (which has it's limitations as explained above), there are tools that make this a really easy task. For example the Steam Web Integration script.
It shows you a green checkmark besides every Steam Store link for titles that you already own and a red cross if you do not own a title, yet. That way one can directly see if something is already owned or not.
I can only recommend using SWI. Not only to make things easier for oneself in order to follow the rules but also to help avoid unnecessary tickets and thus trouble for yourself and the giveaway creator.

You should hide giveaways for the game so this doesn't happen again.

Also, please take the time to familiarize yourself with the rules and proceedings of Steamgifts as outlined in the FAQ and Guidelines. Especially the part where it says:

What should I do if I win a game more than once, or if I win a game that I already purchased?
When this occurs, you should contact the giveaway creator and inform them you will not be accepting or redeeming the gift. Explain the situation, and ask the giveaway creator to request a new giveaway winner. During this process you should not mark the gift as "Received", or reveal the key if the option is available. This will help inform us that you are not accepting the gift, and it will prevent you from being suspended for not activating a gift you won. Once the ticket from the giveaway creator is approved for a new winner, the win will be removed from your profile.