A Positive Approach
Lets start from, We're humans and we're known for making mess but have you thought how to turn that blunder into something affirmative? That too without being apologetic!
Well, the right perspective can help you with that.
Let's say you sent an Email to one of your client which had a slip up in it that you didn't notice and your superior asks you to be more prudent with a hint? What're you gonna say!
"Sorry" comes instinctively from our mouth but do you think saying " Thank you for pointing it out, I'll keep this in mind as this will help me to be more discreet and watchful after this" would be more significant!
Which one of this do you think has a better ring?