In what was a topsy-turvy contest that contained seven goals, County took the lead through Enoch Showunmi's strike, after the forward managed to get on the end of Alan Sheehan's cross.
Then came the game's turning point, as goalscorer Murray was sent off for an off-the-ball incident, which the Posh took advantage of.
Following a few attempts on goal through the returning Lee Tomlin, Perterborough finally got their first courtesy of Ajose, who controlled the ball well before firing past Bartosz Bialkowski to give the hosts a lifeline.
Britt Assombalonga headed home Posh's second to level the contest, before Ajose added his second and his side's third with a strike from outside the box to give the hosts the lead for the first time on the night.
10-man County battled on and levelled the match through Jack Grealish's left-footed shot just two minutes after going behind.
The game didn't end there though, as Ajose completed his hat-trick after heading in from close range past a helpless Bialkowski to signal the end of a hugely entertaining game.