Do I need an undergrad degree in industrial engineering to study a masters in industial engineering?

Given your additional info. . . nope, definitely not.

You should be plenty fine unless the particular school is strangely picky -- I've known several people who've come out of other engineering disciplines to go to some branch of IE.

Your masters degree does not have to match your undergraduate degree but it will make getting accepted much easier. You will probably at least need to have an undergraduate degree in another field of engineering so that you will be taken seriously by engineering masters programs.

Edit: With a degree in Electrical you will be able to do it. You may have to take an extra semester or two of basic undergrad industrial classes but it won't be bad. Just start contacting some grad schools and see what their process is.

It is highly recommended to have an undergraduate degree in some type of engineering or a similar field. Industrial engineering is a very specialized field of study, so if you have an undergraduate degree in something else, like business, you would need to take a lot of additional courses at the graduate level just to "catch up" with the other students. Engineering students take a lot of science and math related courses, like physics, thermodynamics, statics, calculus 1, 2 & 3, etc. If you entered graduate school without these courses, you would be required to take them before you were able to proceed. It's probably best to just get an undergraduate degree in engineering first.