You don't need to prove it. You just need to justify the number you show.

For example: you build a feature to notify customers via email that they have a balance due. The email includes a link with UTM tracking stuff. You also have metrics as to who pays and when. So now you can say "X number of people paid their balance within Y hours of an email being sent out, and within Z minutes of the email linked being clicked" and total up the amount of balances paid. Sure, your feature isn't 100% responsible for enabling that balance to be collected, but it deserves to say it contributed to a portion of the collected balance value.

I wouldn't say "nearly impossible", although big companies do tend to have a few teams in that position. Everywhere I've worked, large or small, there have been a substantial number of engineers who justify their work in terms of money for the company and rocket up the career ladder because of it. (And I've seen a lot of seemingly "cog in the machine" tasks that have large dollar amounts attached as far as management is concerned.)