Upvote:2
The rules for using the card are available fromthis page
https://tfl.gov.uk/campaign/rail-travel-outside-the-london-fare-zones#on-this-page-1
Basically, yes you can even though you are outside the normal area for Oyster services. Make sure you do not get the Gatwick Express but, as you say in your question, get Thameslink.
Check for possible strike action and other disruption on the National Rail site here
http://www.nationalrail.co.uk/service_disruptions/today.aspx