Emoji work in most browsers. The problem is that many fonts are missing the corresponding characters. You just need to pick a good font. You'll be missing full color icons, though.
Example page with an emoji character that works almost everywhere (with a web font based on Symbola): http://unicodepanda.com/
I would guess that most fonts will lack emoji characters, and always will, just like most fonts won't have all umpteen-thousand hanzi/kanji. I'm no expert on HTML fonts, but I thought it was the browser's job to find and use a font which contains the character on the page. I can write Japanese text, and specify it should be "Comic Sans", and every web browser I've ever tried will pick a reasonable font to display them, rather than leaving a Comic Sans-sized blank area on the page.
OS X comes with at least one font with these characters, yet apart from Safari, none of my web browsers can seem to use it. To me, that means it doesn't work.
Example page with an emoji character that works almost everywhere (with a web font based on Symbola): http://unicodepanda.com/