Warrington, who are two divisions below Wealdstone, looked like they had earned a replay when substitute Joseph Piggott levelled the scores up with his first touch in the 83rd minute, but the away side's hearts were broken when Green converted his second of the game with two minutes remaining.
The opening ten minutes of the game were largely even, although Wealdstone goalkeeper Jonathan North had to be alert to come off his line to prevent Warrington's Jamie McDonald from having a clear run at goal.
The away side grew in confidence after that half chance and they had a glorious opportunity to take the lead when the ball was worked to William Hayhurst inside the penalty area, after the Stones carelessly lost possession in their own half, but he could only shoot straight at North from a promising position.
Warrington thought they had taken the lead seconds later when McDonald tapped the ball home from close range after a good save from North, but the flag was raised for offside.
Wealdstone's first effort of the game came just before the 18 minute mark, with Mustafa Tiryaki doing well to flick the ball over the defender's head, but his volley went well wide of the target.
The Stones slowly got back into the game after a nervous opening half an hour, with former Watford man Matthew Whichelow stinging the palms of Warrington goalkeeper Anthony McMillan with a curling free kick.
Wealdstone boss Bobby Wilkinson was clearly disappointed with his team's first half display and he responded by bringing on forward Dan Fitchett for Taofiq Olomowewe in the 35th minute, with midfielder Sam Cox moving to right back.
Wilkinson's change almost had the desired effect just before half-time as after good hold up play from Tiryaki, Green released Fitchett in behind the Warrington defence and the Stones sub forced McMillan into a good save low down.
The Stones started the second half much better, with Tiryaki having a header blocked by a defender and Green seeing a 20-yard drive go over the bar.
But Wealdstone had their goalkeeper to thank five minutes later, as North made a superb finger tip save to deny Warrington's Sean Williams' dipping volley from distance.
The game had started to open up in the second half as both teams went for the win, with Ashleigh Artwell's strike being deflected wide, after good build up play from Wealdstone.
After the resultant corner came to nothing, Warrington went on the counter attack and came inches away from taking the lead, with North making another superb save and Cox brilliantly clearing the ball off the line twice in quick succession to keep the Stones on level terms.
But back came the Stones and in the 65th minute, they were handed a great chance to take the lead, after referee Sam Purkiss instantly pointed to the spot after Eddie Oshodi was adjudged to have been fouled inside the area by a Warrington defender. Green calmly stepped up and slotted the ball into the bottom right hand corner, to put Wealdstone 1-0 up.
Despite going 1-0 down, Warrington were still a threat and North inadvertently saved with his feet after Oshodi made a mess of a clearance.
The visitors continued to cause Wealdstone problems and they missed a glorious chance with just over 10 minutes to play, as Dylan Vassallo met a pinpoint cross at the back post, but his volleyed effort went just over the bar.
But Warrington were to make up for that miss just minutes later through substitute Piggott, who put the visitors back on level terms with his first touch, as he calmly slotted the ball home after being played through on goal by Jack Higgins.
It looked like the game was heading for a replay but there was to be late drama at Grosvenor Vale, as Wealdstone scored an 88th minute winner to keep their dreams of going to Wembley alive. Green picked up the ball outside the area and hit a low effort which was fumbled over the line by the unfortunate McMillan to send the Stones crowd wild and Bobby Wilkinson's men into the next round of the FA Trophy.