Amoo struck back for Carlisle in the 71st minute, driving home right footed from 12 yards after Tom Lawrence had provided the cross.
A minute later he was on target again, side footing substitute Matty Robson's searching ball from the left into the net from close range.
Peterborough missed a late chance to salvage a point when Mark Little was fouled in the box. Goalkeeper Ben Amos got fingertips to Grant McCann's penalty and the ball rocketed off the right-hand post and back into play.
Irons said: "A solid back three provided us with a platform to build on. Their goal was a sucker punch, but the response was magnificent and the fans drove us on.
"David Amoo has been a star for us this season. He has everything in his locker.
"It was an excellent game and we deserved to come away with what was a huge three points for us. It shows the spirit in the side.
"We have had a couple of setbacks, but we are slowly making Carlisle a difficult place to come to. We came back from the Preston defeat with a vengeance and there was only one team that deserved to win."
- Sports & Recreation
- Lee Tomlin
- Matty Robson
- David Amoo