The original problem was the extension used the secure domain, but CloudFlare is set up on the unsecure domain. After the fix, it works great.
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With CloudFlare® control Magento extension you can control main features of CloudFlare® directly from your Magento admin page. CloudFlare® Control extension supports all the main methods of CloudFlare® API. You can save a lot of time working with CloudFlare® settings directly from Magento backend, so this is a must have extension for any Magento store what use CloudFlare®.