Blackstock struck in the 38th minute as a third win in an unbeaten run of four away matches moved the Midlands club seven points clear of the Championship's bottom three despite the 70th-minute dismissal of winger Garath McCleary.
Both teams made just a single change at London Road on Monday, despite a second game in the space of 48 hours.
Teenage midfielder Joe Newell was handed a rare start for Peterborough as forward Lee Tomlin dropped down to the bench.
Guy Moussi was missing from the Forest squad. He was replaced in midfield by Lewis McGugan.
Forest flew out of the blocks. McCleary and Andy Reid had goalbound efforts blocked inside the opening minute and McGugan sent a shot wide before Posh goalkeeper Paul Jones had to tip a Blackstock header over the crossbar in the fifth minute.
McGugan struck a post with a strike from 18 yards two minutes later before Jones was called into action again, this time to make a parrying save to deny Radoslaw Majewski.
Steve Cotterill's side continued to pile forward with Adlene Guedioura and McGugan once again threatening with long-range efforts.
Peterborough were forced into a 16th-minute change when Emile Sinclair suffered an injury so David Ball was introduced.
Following wave after wave of Forest attacks, Newell offered something in reply after 20 minutes with a shot from the edge of the area that cleared the bar. George Boyd followed suit shortly afterwards.
Jones was fortunate to get away with a fumble from a Blackstock header midway through the first half before Blackstock and Majewski saw further efforts narrowly miss the target.
Forest finally made the breakthrough seven minutes before half-time.
McCleary's cross from the right was skilfully brought under control by Blackstock in front of goal and the striker lashed the ball into the net via the underside of the crossbar.
Gaby Zakuani headed over as Peterborough looked for an immediate equaliser but Blackstock would have had a second goal had it not been blocked by Jones.
Posh threatened first after the break with Lee Camp parrying a shot from Paul Taylor.
But the second half soon began to take on the same pattern as the first as Jones pushed an effort from Majewski past a post and Blackstock saw shots blocked by Shaun Brisley and then Zakuani.
However, the sending-off of McCleary would see Forest lose their momentum.
The winger was shown a straight red card following a 70th-minute challenge on home captain Grant McCann.
Peterborough immediately set about trying to make their numerical advantage count and Camp had to block another shot from Taylor.
The Posh striker then saw another goalbound attempt blocked by Forest skipper Luke Chambers.
McCann curled a free-kick wide after 77 minutes and Boyd spurned their last chance to get something from the game when he mis-hit his shot from inside the area with two minutes remaining.