I've just determined using Firebug that when Fancybox window is created it actually takes all of my ASP.NET controls (contained in DIV tag) and puts them outside FORM tag. So I guess this is the reason why then ASP.NET button doesn't do anything - it is placed outside form.
So, do you have any suggestions how I can prevent this (or make that ASP.NET button work), other than using completely different modal dialog?
EDIT: OK, people are reporting that some of the proposed fixes are working for them on certain versions. So, be sure to read all of the answers / scroll to the bottom for how to fix this issue on different Fancybox versions.