Take a particular convoluted piece of code from your codebase, preferably having a bug in it that exists in your issue tracker.
Show your manager this code, by scrolling through it (he/she does not have to understand it). Now you say: "We have this bug in this code, which cost me 4 hrs to solve. I had to debug, consult team members, etc. Now had we refactored this code before, the bug would already be eliminated and never reached the customer. Here I have the refactored code, which took me 2 hrs to create. Looks much cleaner, right? Even if the bug had remained in this code, I would have found and fixed it within an hour, and so would the next guy without having to bother other team members. When I refactored I even found yet another bug in the original code that hadn't been detected yet."
Refactoring turned out to be cheaper in any way, and avoids the hit to customer satisfaction if it goes into production.