Saints took the lead in unusual circumstances when a Kai Wheeler corner crept inside the far post.
Binley turned the game around before half-time with two tap-ins at the far post as Saints struggled to push the opposition up the pitch.
The second half saw the away side take control - Sam Cooper and Josh Sly dominating the midfield. A flurry of missed opportunities to draw level followed - with Binley struggling to get out of their half - before the home side notched a third with their first shot of the half.
Leon Juius bagged a reply for Southam, but as the Saints continued to push for an equaliser, Binley broke clear to score a fourth late on to make the game safe.