No, the Digest Access Authentication Scheme is a little more complex as it implements a challenge-response authentication mechanism that requires the following steps:
- client sends a request for an access-protected resource, but an acceptable Authorization header field is not sent
- server responds with a "401 Unauthorized" status code and a WWW-Authenticate header field (the digest-challenge)
- client sends another request for the same resource but containing a Authorization header field in response to the challenge (the digest-response)
- if the authorization is not successful, go to step 2; otherwise the server proceeds as normal.
This means there are at least two request/response pairs.
Each WWW-Authenticate response header field has the syntax:
challenge        =  "Digest" digest-challenge
digest-challenge  = 1#( realm | [ domain ] | nonce |
                    [ opaque ] |[ stale ] | [ algorithm ] |
                    [ qop-options ] | [auth-param] )
So you need to parse the digest-challenge to get the parameters to be able to generate a digest-reponse for the Authorization request header field with the following syntax:
credentials      = "Digest" digest-response
digest-response  = 1#( username | realm | nonce | digest-uri
                | response | [ algorithm ] | [cnonce] |
                [opaque] | [message-qop] |
                    [nonce-count]  | [auth-param] )
That section does also describe how the digest-response parameters are calculated. In particular, you will probably need an MD5 implementation as that’s the most commonly used algorithm for this authentication scheme.
Here is a simple tokenization that you can start with:
var ws = '(?:(?:\\r\\n)?[ \\t])+',
    token = '(?:[\\x21\\x23-\\x27\\x2A\\x2B\\x2D\\x2E\\x30-\\x39\\x3F\\x41-\\x5A\\x5E-\\x7A\\x7C\\x7E]+)',
    quotedString = '"(?:[\\x00-\\x0B\\x0D-\\x21\\x23-\\x5B\\\\x5D-\\x7F]|'+ws+'|\\\\[\\x00-\\x7F])*"',
    tokenizer = RegExp(token+'(?:=(?:'+quotedString+'|'+token+'))?', 'g');
var tokens = xhr.getResponseHeader("WWW-Authentication").match(tokenizer);
This will turn a WWW-Authenticate header field like:
WWW-Authenticate: Digest
        realm="testrealm@host.com",
        qop="auth,auth-int",
        nonce="dcd98b7102dd2f0e8b11d0f600bfb0c093",
        opaque="5ccc069c403ebaf9f0171e9517f40e41"
into:
['Digest', 'realm="testrealm@host.com"', 'qop="auth,auth-int"', 'nonce="dcd98b7102dd2f0e8b11d0f600bfb0c093"', 'opaque="5ccc069c403ebaf9f0171e9517f40e41"']
Then you need to parse the parameters (check existence and validity) and extract the values. Note that quoted-string values can be folded, so you need to unfold them (see also the use of the unquote function unq in the RFC):
function unq(quotedString) {
    return quotedString.substr(1, quotedString.length-2).replace(/(?:(?:\r\n)?[ \t])+/g, " ");
}
With this you should be able to implement that on your own.