Grant grabbed a second 10 minutes into the second period, and Rovers had only Darren Jones' 81st minute consolation as reward for their diligent efforts to salvage a point.
Lightning fast starts have become a Rovers trademark this season, but even by those standards they were quick out of the blocks last night.
Fleetwood could have opened the scoring inside 10 seconds, surging into the box from the left channel and forcing Shots keeper Nikki Bull into a smothering save.
Forest Green midfielders past and present were instrumental in the opening goal, as Michael Brough robbed Ben Harding on halfway and released Fleetwood again. This time the striker had plenty of time to pick his spot and slide the ball into the bottom corner.
The visitors were being swamped, and it should have been 2-0 on nine minutes when Anthony Tonkin's low cross travelled all the way across to Les Afful on the right side of the area. Perhaps surprised to find himself in such space, the little winger hit his shot straight at Bull.
As with Fleetwood, Grant could have scored moments before his goal when his low shot from range was comfortably fielded by Ryan Robinson.
The Rovers keeper was called into action again when Louis Soares' 20-yard drive had to be clawed around the post, but he was powerless to prevent the 11th minute leveller as Grant rose between Jones and Chris Giles and headed home Soares' cross.
All of a sudden the momentum had switched, and another fine Soares delivery was nodded down by Scott Davies at the back post for Grant to scoop a shot over.
The normally solid Jones was caught dwelling on the ball by Harding in the 21st minute, forcing Robinson to block the former Lawn favourite's effort and Soares' follow-up.
Rovers' luck ran out six minutes later, as they were caught out by Anthony Charles' lofted pass and allowed Dixon to clamber above Robinson to nod into the net. The stopper complained vehemently that he had been fouled, but referee Phil Gibbs broke with modern convention and ignored his protests.
Fleetwood's third clear opening of the half stemmed from Simon Clist's defence-splitting pass, and once again Bull had to react smartly to block with his body.
At the other end, Robinson had to clutch daisy-cutters from Davies and Soares and parry Harding's fierce 20-yarder as indecision lingered in the home defence.
Aldershot were reduced to 10 men four minutes before the break when Afful raced onto Lee Dodgson's pass and was hacked down more in desperation than malice by last man Anthony Straker.
Yet it was Fleetwood who left for half-time with his head in his hands. Afful again exchanged passes with Dodgson and produced an inviting cut-back which was stabbed goalwards by the frontman but instinctively saved by Bull.
The third goal arrived on 55 minutes - ironically as Rovers' numerical advantage was beginning to tell - as substitute Ricky Newman beat Jones to the ball and went wide to Soares, whose deep cross was perfect for Grant to head into the ground and beyond Robinson.
Thereafter it was a night of frustration for the men in black and white, who earn rave reviews for their passing football but could have done with an Alsop-esque target man to batter their way back into the contest.
Dodgson saw a deflected shot saved by Bull and Jonathan Smith blazed over from the edge of the area before Jones benefited from Afful's persistence on the right and notched his second goal of the campaign from 12 yards.
Brough came closest to completing the comeback, diverting Olly James' fine cross wide on the stretch on 84 minutes, while Soares should have added a fourth for Aldershot after getting on the end of Davies' quick break and dragging his shot.
Forest Green: R. Robinson; J. Hardiker, D. Jones, C. Giles (O. James 55), A. Tonkin (J. Smith 46); L. Afful (M. Preece 90), M. Brough, S. Clist, P. Stonehouse; L. Dodgson; S. Fleetwood; Subs not used: J. Pitman, T. Burton.
Aldershot: N. Bull; R. Gier, R. Day, A. Charles, A. Straker; L. Soares, L. Chalmers, B. Harding (D. Winfield 85), S. Davies; J. Grant, J. Dixon (R. Newman 45); Subs not used: R. Elvins, D. Hylton, M. Jaimez-Ruiz.
Referee: P. Gibbs (Birmingham)
Attendance: 1,103
Star man: Stuart Fleetwood - Should have added to his early goal but proved a constant menace to the visitors.
Source: www.gloucestercitizen.co.uk