Blackstock earned his side a passage into the second round with the only goal of the game after 57 minutes at Highbury on Monday night.
The experienced Andy Reid produced a superb delivery from the left which was met by the unmarked striker at the far post.
His first header was well saved by Scott Davies but the rebound fell perfectly into his path and he stroked the ball into the empty net from close range.
A poor first half saw few efforts of any note. After 12 minutes debutant Jon Parkin forced Forest keeper Karl Darlow into a save low to his right after getting in behind his marker to head a Barry Nicholson free-kick low to the keeper's right.
Fleetwood continued on the front foot with a Dean Howell cross just too high for David Ball before the visitors produced their only shot on target in the first 45 minutes with a Greg Halford effort well blocked by Youl Mawene.
But Blackstock settled the tie just under an hour into proceedings when he beat Davies at the second attempt.
Forest went close to a second with Marcus Tudgay forcing Davies into two saves and Lewis McGugan flashing a shot just wide of the right post before Fleetwood staged a late rally.
Substitute Steven Gillespie drilled an effort just over with a minute to go and Darlow reacted well to save a last minute header from Stephen McNulty.