Provided you are on the 2600hz hosted environment, they offer a Call Recording Monster App.
Alternatively, you can use URL based recording with a PHP script but that's more complicated.
Here is a link to a free script I created https://github.com/fastdevice look under Call-Recording.
@Jack Noe I tested Webhook trigger "channel_create" and it appears to work for this purpose.
The best way to investigate the contents of a Webhook is to use a service called RequestBin. (please google and find the RequestBin address) Follow the directions on the site and create a RequestBin, the service will return a URL to be inserted in the Webhook. I like to set the Request Type to POST.
Once you have the Webhook configured with the RequestBin URL, call your callflow and then look at your bin. You should see all the detail from the Webhook appear in the bin. What you do with that data is now up to you, but this is the first step of the process.