During last week's Commons culture, media and sport committee hearing with News Corporation bosses Rupert and James Murdoch, Mensch said that Morgan had been "very open about his personal use of phone hacking" in memoir The Insider.
However, she now admits that she made a mistake and had actually misread a report in The Daily Telegraph about the book. Morgan had consistently denied the claims and demanded an apology from Mensch.
Mensch and Morgan rowed on Twitter after Mensch stood by her remarks, which were made in parliament where comments have more protection from libel proceedings.
The Corby MP - known as the author Louise Bagshawe until she married Metallica manager Peter Mensch in June - asked the select committee to call Morgan to give evidence on phone hacking.
At the time, she said: "As a former editor of the Daily Mirror, he said in his book The Insider recently that that 'little trick' of entering a 'standard four digit code' will allow 'anyone' to call a number and 'hear all your messages'.
"In that book, he boasted that using that 'little trick' enabled him to win scoop of the year on a story about [former England football manager] Sven-Goran Eriksson. That is a former editor of the Daily Mirror being very open about his personal use of phone hacking."
Morgan, who now hosts his own show on US network CNN, called the claim "a complete outrage" and called on Mensch to supply evidence. He also accused the MP of being "cowardly" by hiding behind parliamentary privilege to make the claims.
Sly Bailey, chief executive of Daily Mirror publisher Trinity Mirror, also wrote to the committee on July 21 demanding the MP correct her remarks.
"On two occasions Mrs Mensch made uncaveated statements purporting to be facts," Bailey wrote. "On both occasions she made allegations that are wholly untrue."
Writing to culture committee chairman John Whittingdale today, Mensch said that she had "wrongly stated" that Morgan had "been open about personally hacking phones".
"This was based on my misreading of an article in The Daily Telegraph... which covered Mr Morgan's description in his book of how to hack a phone and how he won the 'Scoop of the Year' on the story of Sven-Goran Eriksson and [TV presenter] Ulrika Johnson," she wrote.
"The Telegraph report covers the claim of a blogger that this story was acquired by phone hacking, and I misread that as Mr Morgan himself claiming this to be true. Therefore, I must apologise to Mr Morgan and the committee for this error about his book."
Writing on Twitter, Morgan said: "Apology graciously accepted @LouiseMensch - thank you."
Last night, another spat erupted between Morgan and Channel 4 News presenter Jon Snow, after the latter wrongly reported a fake tweet saying that Morgan had been suspended by CNN.
Morgan subsequently tweeted: "Sorry to disappoint you all, but I'm afraid poor old @jonsnowC4 got duped by a fake Twitter account. I've not been suspended by CNN."
Later, he added: "Jon snow suspended from Tweeting Piers Morgan henceforth..dupe was duped alas..apologies old chap..sorry, I mean young man!"
Earlier in the week, Morgan was again forced to deny claims linking him to phone hacking after fresh doubts emerged over comments he made in a 2009 interview on BBC Radio 4's Desert Island Discs.