"cross-browser" is a misnomer that seems to have been invented by DOM library authors as a marketing tool.

If we're talking about the en vogue subset of browsers, then QSA has support. If we're talking about a set of browsers where a usable DOM implementation is provided (most dating back to Netscape), then, no, support is not provided.

"Works in all browsers" should be closer to the latter, not the former.