Having gone behind through Macauley Gillesphey's free-kick after Jason Kennedy had been hauled down, Orient overturned the deficit.
Jay Simpson found space to level after half-time, firing into the left corner.
Ollie Palmer then streaked clear to put Orient ahead, before Raynes headed in from Alex Gillead's cross to level.
The goal denied Kevin Nolan's visitors the chance to go fifth in the table, as they remain seventh, while Carlisle are three points behind in 11th.
Leyton Orient player-manager Kevin Nolan told BBC Radio London:
"You've got to respect the point, especially when you come to a place like this.
"In the end we probably could have snatched it, but if we would have I think (Carlisle manager) Keith Curle would have been equally disappointed.
"I'm delighted with the response. Since I've come in we've taken 16 points out of eight games so it's fantastic form and we've got to take positives out of it because we're all disappointed at the moment."
Michael Raynes' late equaliser earned Carlisle United a key point in a 2-2 draw with fellow League Two promotion hopefuls Leyton Orient.