Callum Willock, back after a lengthy injury absence, gave Posh a dream start, storming through the middle on to a second Clive Platt flick-on and moving right before coolly slotting a low shot past Boaz Myhill.
Former Fulham frontman Willock almost struck again on five minutes, bringing a diving save from Myhill, but before the half hour mark Willock limped off with a recurrence of his hamstring problem.
By then the scores were level with Posh caught napping at the back as a quickly taken 22nd minute free-kick was met by top scorer Elliott who got in front of the defender Simon Rae to head his sixth goal of the season.
Bursting forward from midfield, Elliott was denied by the foot on the post after beating Mark Tyler in the last minute before the break, and the Northern Ireland international brought an impressive save from Tyler as the scoring period opened.
Hull maintained the momentum and grabbed the lead on 52 minutes, lanky defender Cort heading across goal and into the net after Tyler appeared to mis-judge the flight of a Michael Keane corner.
The points were secured on 65 minutes, Elliot making no mistake with a low shot after being left clear by an inch perfect cross from Ryan France.
Posh boss Barry Fry reacted by sending on his remaining substitute Curtis Woodhouse and Ryan Semple and switched to four at the back, but a second home goal, nodded in by Andy Clarke from close range in the last of the three added minutes was too little, too late.