When I click a download link pointing to an EXE file, how do I run (ie. Open) it directly?

I know this is possible because I have another computer that does allow it, without additional plugins. The only difference is that the other computer started out with a 2yo version of FF and this computer was re-installed a year ago so it started out with a newer FF. Both have been constantly updated to the newest version. With the 2yo version I have always been able to run EXE files throughout the lifetime.