Green picked up the ball on 72 minutes and clinically finished beyond Tony Roberts as the Silkmen picked up three precious points in their fight for survival.
An almost gale-force wind meant that the home side were contained in their own half for the opening 45 minutes as Macclesfield opted to hit long balls into the box at every opportunity.
It almost paid off when Gareth Evans had an effort chalked off after Richard Walker was adjudged to have fouled Roberts.
Despite being forced to spend their time defending, it was the Daggers who should have gone ahead when the unmarked Jon Nurse headed over from Peter Gain's corner.
Jamie Tolley finally tested Roberts with a straight-forward effort from 20 yards, while Evans' shot was then bravely blocked by Anwar Uddin.
The Daggers got their first shot on goal as Gain's tame drive was straight at Jon Brain, before Danny Foster drilled wide from a tight angle.
Evans came within inches of giving the Silkmen the lead on 40 minutes, but his 25-yard snapshot flew wide.
It was almost 1-0 before the break when Ryan Cresswell came close to putting through his own goal, leaving Brain to berate his shell-shocked defence.
After half-time the Daggers had the wind in their favour and Mark Arber came close from two corners.
The home team had a glorious chance to open the scoring on 57 minutes when James Jennings brought down Nurse in the area, but Dave Rainford's spot-kick was saved by Brain.
After Green broke the deadlock, the Daggers almost got back into the game after a long Jon Boardman cross was fumbled by Brain, but substitute Ryan Hall slid the loose ball into the side-netting.
Brain then repelled everything that the Daggers could throw at him by tipping over Nurse's fierce shot and then blocking substitute Chris Moore's effort as Town got a vital win.