I haven't tried this but you might be able to do it with jQuery.
$('input[type=file]:first').trigger('click')
The reason you can't just do .click() is that this is only supported for inputs of type button.
[EDIT] Ordinarily I'd test this before posting, but I've got to run and pick up my daughter. If it doesn't work, leave a comment and I'll delete my answer.
[EDIT] The following works in Safari, but not Firefox. Didn't test IE as not working in Firefox is probably a deal breaker. Probably need to find a different way of doing it.
<html>
<head>
<script type="text/javascript" src="jquery-1.2.6.js"></script>
<script type="text/javascript">
function clickFileUpload() {
$('input[type=file]:first').trigger('click');
}
</script>
</head>
<body>
<form action="test_submit" method="POST" accept-charset="utf-8">
<p>
<input type="file">
</p>
<p>
<input type="button" onclick="clickFileUpload();" value="Get File" />
</p>
</form>
</body>