Welcome to Android Central! Auto-Sync needs to be on in order for a contact entered into your Contacts app to be synced to your Google Contacts in the cloud -- but this is assuming you saved the contact to your Google account, and not the local Phone account (or any other account, like Samsung).
Also, you may want to go to Settings>Accounts-Google, tap your account, and make sure the individual sync switch for Contacts is turned on.