Here is an answer that is more idomatic for the latest JavaScript.

import AWS from 'aws-sdk';

const invokeLambda = (lambda, params) => new Promise((resolve, reject) => {
  lambda.invoke(params, (error, data) => {
    if (error) {
      reject(error);
    } else {
      resolve(data);
    }
  });
});

const main = async () => {

  // You shouldn't hard-code your keys in production! 
  // http://docs.aws.amazon.com/AWSJavaScriptSDK/guide/node-configuring.html
  AWS.config.update({ 
    accessKeyId: 'AWSAccessKeyId', 
    secretAccessKey: 'AWSAccessKeySecret', 
    region: 'eu-west-1',
  });

  const lambda = new AWS.Lambda();

  const params = {
    FunctionName: 'my-lambda-function', 
    Payload: JSON.stringify({
      'x': 1, 
      'y': 2,
      'z': 3,
    }),
  };

  const result = await invokeLambda(lambda, params);

  console.log('Success!');
  console.log(result);
};

main().catch(error => console.error(error));

Update

Rejoice! The AWS SDK now supports promises:

import AWS from 'aws-sdk';

const main = async () => {

  // You shouldn't hard-code your keys in production! 
  // http://docs.aws.amazon.com/AWSJavaScriptSDK/guide/node-configuring.html
  AWS.config.update({ 
    accessKeyId: 'AWSAccessKeyId', 
    secretAccessKey: 'AWSAccessKeySecret', 
    region: 'eu-west-1',
  });

  const params = {
    FunctionName: 'my-lambda-function', 
    Payload: JSON.stringify({
      'x': 1, 
      'y': 2,
      'z': 3,
    }),
  };

  const result = await (new AWS.Lambda().invoke(params).promise());

  console.log('Success!');
  console.log(result);
};

main().catch(error => console.error(error));