API Gateway、Cognito、IAMの設定はクラスメソッドさんの『Amazon API Gateway の API を Cognito で認証して呼び出す』を参考にしてください

検証バージョン

Node.js 0.10.41
npm
- aws-sdk 2.4.4
- aws4 1.41

ソースコード

AWSのregion、CognitoのIdentityPoolId、API Gatewayのhostname、pathは環境に応じて変更してください

var https = require('https')
  , aws4 = require('aws4')
  , AWS = require('aws-sdk')
  , region = 'ap-northeast-1'
  ;

var cognitoOptions = {
  IdentityPoolId: 'YOUR_COGNITO_IDENTITY_POOL_ID'
};
var httpOptions = {
  service: 'execute-api',
  region: region,
  hostname: 'YOUR_API_ID.execute-api.ap-northeast-1.amazonaws.com',
  path: '/prod/resorucePath',
  method: 'GET'
};

getCredentials(cognitoOptions, function (error, credentials) {
  request(httpOptions, credentials, function (error, body) {
    console.log('body: ', body);
  });
});

function getCredentials(options, done) {
  AWS.config.region = region;
  var cognitoidentity = new AWS.CognitoIdentity();
  cognitoidentity.getId(options, function (err, objectHavingIdentityId) {
    if (err) return done(err);
    cognitoidentity.getCredentialsForIdentity(objectHavingIdentityId, function (err, data) {
      if (err) return done(err);
      var credentials = {
        accessKeyId: data.Credentials.AccessKeyId,
        secretAccessKey: data.Credentials.SecretKey,
        sessionToken: data.Credentials.SessionToken
      };
      done(null, credentials);
    });
  });
}

function request(options, credentials, done) {
  var signedOptions = aws4.sign(options, credentials);
  var req = https.request(signedOptions, function (res) {
    var body = '';
    console.log('statusCode: ', res.statusCode);
    console.log('headers: ', res.headers);
    res.setEncoding('utf8');
    res.on('data', function (chunk) {
      body += chunk;
    });
    res.on('end', function (res) {
      done(null, body);
    });
  });
  req.on('error', function (error) {
    done(error);
  });
  req.end();
}

実行結果

statusCode:  200
headers:  { 'content-type': 'application/json',
  'content-length': '6',
  connection: 'keep-alive',
  date: 'Sat, 02 Jul 2016 00:42:07 GMT',
  'access-control-allow-origin': '*',
  'x-amzn-requestid': 'cd98a1d5-3fed-11e6-b0fe-e73602f4978c',
  'x-cache': 'Miss from cloudfront',
  via: '1.1 7f150809b0d3f0320b40d49f5756b015.cloudfront.net (CloudFront)',
  'x-amz-cf-id': 'CEebYad9DSFzmLrsPKwxLyCaGhgSXr4z4VniqNr9oJzVcXo9DuFzrQ==' }
body:  "hoge"

参考資料