Apple only has one method to backup and restore data on your device, and it is through the iTunes backup feature. This is done using a computer to backup the information from your device. If you are using iCloud to backup your data, there is no guarantee that your game statistics and data will be backed up. However, a device backup would be the only method to transfer game statistics.
If your game data was included in your most recent backup, you should be able to restore from your old device to your new device. Once the restore setup of your iPhone is complete, your data should be restored to your new device with your statistics intact. Unfortunately, this process is the only method possible at this time.
Once backed up and restored, you can log into Game Center on your new device to continue using the same username and password as used on your old device.
You can visit the Apple Support page for additional details about Apple backup feature: https://support.apple.com/en-us/HT204136