The League One top scorers attacked from the very start and it took them only four minutes to go in front, Posh amazingly leaving Hayter totally unmarked to nod in a right wing cross form Wade Elliot for his 14th goal of the campaign.
Fast advancing home goalkeeper Mark Tyler managed to halt Elliot's solo run as Bournemouth continued to press and Hayter, also the division's top individual scorer, and Karl Broadhurst, each had efforts during a frantic goalmouth scramble as Posh struggled to clear their lines.
Posh showed some attacking spirit before half-time, Bournemouth needing a fine tackle from skipper Broadhurst to stop Richard Logan's break and Clive Platt put in a couple of half decent headers before veteran Any Legg was only just to high with a long-range shot.
For the second half Posh replaced Peter Kennedy and Danny Sonner with David Farrell and Gareth Jelleyman and switched to a more attacking formation although it was defender Sagi Burton who came closest to scoring with a blistering shot from outside the area that brought a superb one handed save from Bournemouth goalkeeper Neil Moss.
With Bournemouth increasingly happy to fall back and protect their lead, referee Trevor kettle turned down loud home penalty appeals after Platt was bundled over by Martin Cranie and then Posh boss Barry Fry emphasised his attacking commitment sending on veteran striker Andy Clarke.
But Posh were unable to turn pressure into clear cut chances and both Platt and skipper Curtis Woodhouse failed to exploit a dangerous situation set-up by Adam Newton's fine right sided break and accurate cross in the closing minutes.