Brown, who goes straight into Saturday's Emirates FA Cup squad to face Basingstoke, joins on a one month loan from the Hammers.
The pacey 18-year-old joined West Ham from Arsenal in 2013, scoring seven goals in the 2013/14 campaign for the Hammers' Under-18s.
He then moved into the club's Development Squad where he netted three times.
The Clarets would like to extend a warm welcome to Jordan, and wish him a successful and enjoyable stay at the Melbourne Stadium.
Updated 09:53 - 10 Oct 2015 by Adam May