Southam, who signed for the Clarets from last season's Conference South Champions Eastleigh, struck a low shot underneath the body of Jack Giddens on 28 minutes.
Goals from two trialists in the second-half, one a sublime shot from just inside the area and the other a tap-in from 4-yards following Joe Ward's strike, were enough for the Clarets.
The next pre-season friendly for City is on Monday July 14th at home to Grays Athletic.
www.chelmsfordcityfc.com will be providing a weekly round-up on the pre-season matches.
@OfficialClarets, meanwhile, will be providing goal updates as and when they happen.