Colby Wood was on target to earn Heacham their first win of the season and lift them off the foot of the standings.
After nine consecutive defeats, a 1-0 win at fellow strugglers Cornard United was enough to propel the Magpies up the Thurlow Nunn Premier Division table.
They have chance to build on that next Saturday when they welcome new bottom-placed side Kirkley & Pakefield to Station Road.
In the senior section of the Anglian Combination, Castle Acre Swifts remain without a point in the Premier Division after going down 4-1 at home to Holt United.
Morning leaders Watlington lost 3-1 at Fakenham Town Reserves in Division One, where Thomas Wagg had given them hope with a first-half goal.
Dersingham Rovers bounced back from their opening day defeat in stunning fashion with a 7-1 romp against visitors Caister.
In the Cambs League, Outwell Swifts enjoyed a 4-0 victory over Longstanton in Senior B Division.
