[June 2020: The previous answer has been replaced due to it returning incorrect results].

Most modern JS environments (browsers and Node) now support the TextEncoder API, which may be used as follows to count UTF8 bytes:

const textEncoder = new TextEncoder();

textEncoder.encode('⤀⦀⨀').length; // => 9

This is not quite as fast as the getUTF8Length() function mentioned in other answers, below, but should suffice for all but the most demanding use cases. Moreover, it has the benefit of leveraging a standard API that is well-tested, well-maintained, and portable.