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."

UK universities also do well lower down the tables, with 18 in the top 100 and 10 from London alone in the top 400.
John O'Leary, part of the board that compiled the latest ranking and editor of the Times Good University Guide, says a greater emphasis on research has helped make world-beaters of a handful of UK universities.
Mr O'Leary told BBC News: "UK universities are improving the effectiveness of their research. All but three of the UK's top universities achieved higher citation rates than last year, though they still trail the big US institutions like Harvard, Stanford and Massachusetts Institute of Technology.
"There are now three sets of international rankings for universities - all of them use citations among their main measures. Leading universities are increasingly aware of their international position and academics always want to produce research that is cited by others. American academics tend to cite other Americans, which gives them an edge.
"Increasing the number of academic citations is one of the ways that UK universities have improved this year, but they also do very well in most of the other measures including their reputations amongst other academics.
"UK universities genuinely have a very high reputation all over the world."
Cambridge University, now in second place, was top of the whole table in the two preceding years. University College London (UCL) rose from seventh to fourth place within the past year.
Prof Malcolm Grant, provost of UCL, said the ranking would help confirm London's position as a global leader in something other than banking.
"Higher education, science, research, technology and health are all remarkable strengths of London, every bit as strong as financial services," he said.
"What is remarkable for London is to be the only city in the world to have two top 10 universities.
Mr O'Leary said UCL in particular had done a lot better than in previous years in terms of the number of times its research had been cited in academic journals.
He said UCL's increasing focus on research followed a massive increase in the number of postgraduates at the university in recent years.
In 1995-6 there were 4,270 graduate students at UCL. By 2011-12 that number had grown to 11,000, a move Mr O'Leary said would have brought in more international students and boosted both the university's income and the effectiveness of its research.
Mr O'Leary said teaching in English also gave UK universities a big advantage, particularly in attracting international students. He added: "Per dollar, the UK has for sometime had the most productive university system in the world."
Dr Wendy Piatt, director general of the Russell Group of research universities, said: "It is important to nurture leading research universities within a genuinely diverse education system where a range of institutions with different missions and strengths are fostered.
"The country does not need - and certainly cannot afford - all universities to be research-intensive.
"In these difficult economic times, our institutions have made every pound work hard. Against the odds, with 1% of the world's population, 14% of the world's most highly-cited papers come from UK-based research.
"We can strengthen this effect by prioritising research funding where it will have the most impact."
Michael Raynes' late equaliser earned Carlisle United a key point in a 2-2 draw with fellow League Two promotion hopefuls Leyton Orient.
[NEXT_CONCEPT]
The United Kingdom punches above its weight in terms of the international reputation of its universities, with four in the top six of the latest rankings from QS.