"One of the questions we always asked everyone was "What's the biggest mistake you've ever made in production?""

>>>>  This is my favourite question to ask interviewees.  I mention an example mistake from my past as part of the question.
Denial that you have made any mistakes in production most likely means you: 1) have extremely limited production experience,
2) have very small task lists with no deadlines, [this is a variation on #1]    or 3) are completely unacquainted with truth and cannot face reality. 
How we take ownership of and respond to our mistakes is a test of character as well as technical skill. 
Ideally, we admit our mistake(s), have some documentation of what we did, and have some idea on how to repair the damage.
Depending on the situation, we might need to create a backup of the disaster, before attempting corrective action.
Also, depending on the impact and scope of the mistake, communication with multiple levels of management and the business might be required.
At a minimum, tell your manager and your immediate team.  Hopefully your company culture has the wherewithal to effectively deal with mistakes.
(Firing is not effective.  Mistakes, by definition, are not done with malice aforethought.)
For  extremely small mistakes, you might be able to fix it before the whole company knows about it, but no guarantee.
The ripple effect can be amplified into a  tsunami.

Proceed with caution!!!  Assumptions are required, but one or more of them are what led to the mistake in the first place. 
Verbalize, document and acknowledge your assumptions.
You might find that you're assuming your last step was the only mistake, when the larger mistake occurred three or nine steps earlier.
Stop.  Assess.  Communicate.  Discuss.  Create a restore point if possible.  Assess.  Take corrective action.
If this post sounds like I have made mistakes, then it accurately reflects reality.  I have had a little success in not repeating some of the more painful mistakes, though.

The humility of admitting it is possible for us to make mistakes, regardless of our experience and knowledge, is a virtue to be practised every day.