Lee Tomlin gave Posh a second-half lead after Hogan Ephraim had been sent off. However, Amoo's quick-fire double turned the game on its head and there was more drama to come as Grant McCann missed a late penalty for the visitors.
Although the first half remained goalless, there was no lack of goalmouth action and there were some meaty challenges flying about in midfield adding to the competitive nature of the game which was being watched from the Brunton Park directors' box by Sir Alex Ferguson - father of Posh boss Darren.
Peterborough went down to 10 men in the 52nd minute when on-loan QPR midfielder Ephraim was shown a straight red card for a foul on Liam Noble.
But Posh grabbed the lead nine minutes later after Sean O'Hanlon had been issued with a yellow card for a foul on Tomlin five yards outside the area. Tomlin took the free-kick, curling a shot over the wall and beating Ben Amos as the ball bounced in off the underside of the crossbar.
Amoo struck back for Carlisle in the 72nd minute, driving home a right-foot shot from 12 yards out after Tom Lawrence had provided the cross.
And 70 seconds later he was on target again, side-footing substitute Matty Robson's searching ball from the left into the back of the net from close range.
Peterborough missed a late chance to salvage a point when Mark Little was fouled inside the box. Amos got fingertips to McCann's penalty and the ball rocketed clear off the right-hand post.