SELECT NOW(3)

does the trick.

Javascript timestamps are based on the venerable UNIX time_t data type, and show the number of milliseconds since 1970-01-01 00:00:00 UTC.

This expression gets the current time as a Javascript timestamp integer. (It does so correctly regardless of the current time_zone setting.)

 ROUND(UNIX_TIMESTAMP(NOW(3)) * 1000.0, 0)

If you have TIMESTAMP values stored in a column, you can retrieve them as integer Javascript timestamps using the UNIX_TIMESTAMP() function.

  SELECT ROUND(UNIX_TIMESTAMP(column) * 1000.0, 0)

If your column contains DATETIME columns and you retrieve them as Javascript timestamps, those timestamps will be offset by the time zone offset of the time zone they're stored in.

CREATE TABLE times (
     dt DATETIME(3), 
     ts TIMESTAMP(3)
 );

makes a table with millisecond-precision date / time fields.

 INSERT INTO times VALUES (NOW(3), NOW(3));

inserts a row containing NOW() values with millisecond precision into the table.

 INSERT INTO times VALUES ('2015-01-01 16:34:00.123','2015-01-01 16:34:00.128');

inserts specific millisecond precision values.

Notice that you must use NOW(3) rather than NOW() if you use that function to insert high-precision time values.

%f is the fractional precision format specifier for the DATE_FORMAT() function.

SELECT DATE_FORMAT(NOW(3), '%Y-%m-%d %H:%i:%s.%f')

displays a value like 2016-11-19 09:52:53.248000 with fractional microseconds. Because we used NOW(3), the final three digits in the fraction are 0.

If you have a Javascript timestamp value, for example 1478960868932, you can convert that to a MySQL fractional time value like this:

FROM_UNIXTIME(1478960868932 * 0.001)

It's simple to use that kind of expression to store your Javascript timestamp into a MySQL table. Do this:

INSERT INTO table (col) VALUES (FROM_UNIXTIME(1478960868932 * 0.001))

(Obviously, you'll want to insert other columns.)