The Reds went into the game following back-to-back defeats to Ipswich and Hull, but Blackstock, who last scored at the City Ground in a 2-0 win over Portsmouth at the end of April, set them on their way to victory with a header after 59 minutes.
His strike partner Billy Sharp sealed all three points in the 76th minute, keeping his cool after he was put through on goal by Andy Reid.
Charlie Austin had an early opportunity. The Clarets striker found space down the left side of the penalty box, but showed no sign of the composure that has seen him score 21 times so far this season and his effort joined the Burnley fans in the Bridgford End.
The visitors were having a lot of joy down the right and Martin Paterson's shot clipped the outside of a post midway through the first half following Kieran Trippier's deep cross.
There was another close shave for Sean O'Driscoll's side when Chris McCann fired narrowly wide. Then, a minute before half-time, Forest goalkeeper Lee Camp did well to save Martin Paterson's deflected effort.
O'Driscoll would have been relieved to get his players into the dressing room still on level terms and the hosts were much improved after the break.
Lewis McGugan volleyed over from Reid's cross early in the second half before Blackstock broke the deadlock. The striker had a lot of work to do with Reid's deep cross but he leapt brilliantly to loop his header over Lee Grant.
It took a superb save from Camp to preserve the lead as Burnley sought an immediate reply. The Northern Ireland international got down well to his left to block when Austin seemed certain to score.
But Sharp sealed the victory following good defensive work by Alan Hutton. The on-loan Aston Villa full-back sparked a swift Forest counter attack and Reid slipped in Sharp behind the Burnley defence and he finished confidently when one-on-one with Grant.