I would start by reading this book by Jessie James Garrett: http://www.jjg.net/elements/
It is regarded as the source of user experience practices based on when websites were booming and the updated version attempts to show you that the higher level abstract ideas still carry through to today. There are two starting points you can use to show the value of UX practices:
Starting point A
Business needs, user wants; Developers create value in a product, DevOps allow you to deliver that value to your users and customers and UX enables us to guide the development so that users can access that value. This is more of an abstract starting point but you need to take into account your business's view and essentially the bottom line. Does it make us more money? Does it help us make decisions faster? Being able to answer these questions with examples will help you on your quest.
Starting point B
Look at your organisation's current practices (for example I work using Agile) and see how you can fit it in to augment and provide value to the process. I won't lie and say it will be easy, but you will most likely have to present a use case and do some work in your personal time. Take an existing project in your organisation, preferably one you're currently working on, and use these techniques to sketch, prototype, research and back up the claim that "UX will help us" with hard facts and data.
As much as we can see and feel that it will bring value, you will need to convince your colleagues, senior team and director team if you truly believe that you can improve the current workflow.
These will help if your organisation is open to changes and suggestions. When push comes to shove though you may need to think about switching organisation if you want to use it a lot more in your day-to-day work. If the entire organisation doesn't envisage embracing UX principles, then what's the point?
Best of luck!
Sources: DevOps Engineer and MSc in HCI