The Lancashire club had hoped to mark their first ever appearance in the competition – and their first competitive fixture as a Football League side – with a famous cup victory at The Highbury Stadium.
But Blackstock’s second half strike ensured Sean O’Driscoll’s career as Forest manager got off to a successful start.
The experienced front-man bagged the game’s only goal 12 minutes after the break to stave off the threat of a shock.
O’Driscoll gave debuts to four of his summer signings in Danny Collins, who captained the side, Dan Harding, Greg Halford and Daniel Ayala.
He also handed Karl Darlow his first start for The Reds with the 21 year old goalkeeper deputising for Lee Camp, who was away on international duty with Northern Ireland.
Darlow was forced into the action for the first time after 12 minutes as he made a regulation save from Jon Parkin’s glancing header.
Forest, who lined-up with a 4-4-1-1 formation, struggled to fashion first half opportunities on the Fylde coast.
Radoslaw Majewski squandered arguably their best opportunity of the opening 45 minutes as he volleyed over from a difficult angle after Greg Halford’s initial effort had been blocked.
Forest made a brighter start to the second half in deteriorating weather conditions and Halford was just inches away from getting on the end of a Lewis McGugan free-kick whipped in from the right.
The woodwork came to Fleetwood’s rescue as O’Driscoll’s men went in search of a breakthrough goal. Blackstock got on the end of McGugan’s pin-point cross from the left but his touch came back off the bar.
Home goalkeeper Scott Davies made a smart save moments later from Adlene Guedioura’s fiercely struck shot from a difficult angle.
But The Reds’ travelling support did not have to wait too much longer to celebrate a goal as Blackstock opened his account for the season.
Reid fired over an appetising cross from the left and although Blackstock, who had escaped the attention of his marker, saw his initial header parried by Davies, he reacted quickest and fired home the rebound.
With Forest looking to build on their lead, Davies spilled a long-range free-kick from Reid but his blushes were spared as the visitors failed to capitalise on his slip.
Forest’s growing confidence was exemplified by a spectacular overhead kick by Halford from a partially cleared corner but Davies made a comfortable save.
McGugan exerted a growing influence on the game as it wore on and he had a chance to double the lead 16 minutes from time but fired across the face of goal after a typically surging run from midfield. Blackstock was inches away from getting on the end of a Guedioura cross from the right as he went in search of his second of the night.
The goal scorer was replaced nine minutes from time by Marcus Tudgay and the substitute went close to marking his arrival in style as he headed narrowly wide from another perfectly flighted Reid cross.
The pair combined again five minutes from time – Reid sending over a corner from the right and Tudgay heading straight at Davies.
Fleetwood almost had the last word with substitute Steven Gillespie firing inches over from the edge of the area but The Reds secured their passage through to the next round.
Forest: Darlow, Moloney, Ayala, Collins (c), Harding; Halford, McGugan, Guedioura, Reid; Majewski (Greening 88'); Blackstock (Tudgay 82').
Subs not used: Evtimov, Lascelles, McGoldrick, Miller, Derbyshire.
Scorer: Blackstock 57'
Fleetwood: Davies, Beeley, Howell, McNulty(c), Parkin, Nicholson, Mangan, McLaughlin (Fowler 69'), McGuire, Ball (Gillespie 62'), Mawene.
Subs not used: Milligan, Brown, Goodall, Kinsella, Thewlis.
Bookings: McGuire 73'
Attendance: 3,611 (Away: 1,060)
Referee: M Naylor, South Yorkshire