It turns out this was easy, but it was hard to find complete documentation. Here is how to make a request with request (http://npmjs.org/package/request) and unpack a gzipped body.

gzip_test.js

var request = require('request');
var zlib = require('zlib');

var url = "http://api.stackexchange.com/2.1/search?order=desc&sort=activity&intitle=backbone&site=stackoverflow";

request({
    encoding: null, // this keeps the body as a buffer and not a string
    url: url,
    headers: { 'accept-encoding': 'gzip,deflate' }
  }, function (err, res, body) {
    if (err) return console.log('error', err, res, body)

    // send the body stream to be unzipped.
    zlib.unzip(body, function (err, buffer) {
      if (err) return console.log('error', err)

      // log the unzipped body to
      console.log(buffer.toString())
    })
});