The wisdom of the anonymous monk led me on a quest to seek out the mysteries of Win32::API. However, I couldn't seem to make that work, but luckily I stumbled upon Win32::Shell, which solved all my problems:

use Win32::Shell;

Win32::Shell::Execute("open", "http://www.perlmonks.org", undef, undef
+, "SW_SHOWNORMAL");
[download]

It started the default browser opened to the correct URL, with no delay, and no console window.

Do I get extra XP for answering my own question? :)