This is working for me.

Object.defineProperty(navigator, 'userAgent', {
    get: function () { return 'Mozilla/5.0 (Windows NT 6.2; WOW64; rv:28.0) Gecko/20100101 Firefox/28.0)'; }
});

It is an updated version of code4coffee's answer as Object.prototype.__defineGetter__() is deprecated: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/__defineGetter__

You can programmatically do this in Javascript (this example mocks up Firefox):

navigator.__defineGetter__('userAgent', function () {
    return "Mozilla/5.0 (Windows NT 6.2; WOW64; rv:28.0) Gecko/20100101 Firefox/28.0)"
});
navigator.__defineGetter__('appName', function () {
    return "Netscape"
});

You can then view the changes in the console via (and of course check these via Javascript):

navigator.userAgent
navigator.appName

Here's an example of a test that should work (using Jasmine):

describe("isUserAgentInternetExplorer", function () {
    it("should return false for Firefox", function () {
        navigator.__defineGetter__('userAgent', function () {
            return "Mozilla/5.0 (Windows NT 6.2; WOW64; rv:28.0) Gecko/20100101 Firefox/28.0)"
        });
        navigator.__defineGetter__('appName', function () {
            return "Netscape"
        });
        //your code here...
        expect(...your code here...).toEqual(false);
    });
});

2022

You are not allowed to modify the following headers, according to Fetch WHATWG spec.
And in 2022 all the evergreen browsers have implemented this requirement.

`Accept-Charset`
`Accept-Encoding`
`Access-Control-Request-Headers`
`Access-Control-Request-Method`
`Connection`
`Content-Length`
`Cookie`
`Cookie2`
`Date`
`DNT`
`Expect`
`Host`
`Keep-Alive`
`Origin`
`Referer`
`TE`
`Trailer`
`Transfer-Encoding`
`Upgrade`
`Via`

As well as headers, starting with proxy- or sec-.

These are forbidden so the user agent remains in full control over them.

2024

The User-Agent header is not forbidden anymore:

The User-Agent header is no longer forbidden, as per spec — see forbidden header name list (this was implemented in Firefox 43) — it can now be set in a Fetch Headers object, or with the setRequestHeader() method of XMLHttpRequest. However, Chrome will silently drop the header from Fetch requests (see Chromium bug 571722)

If you are using a XMLHttpRequest you can set a custom Request Header like:

var xhr = new XMLHttpRequest(...);
xhr.setRequestHeader("User-Agent","test");