The email revealed a clear bias toward the government which undermines the integrity of the ABC as an independent broadcaster.
Following the email revelation, more than 100 staff gathered outside the national broadcaster's Sydney headquarters to back a union motion calling for Milne to stand down.
After a meeting on Thursday, the ABC Board released a statement saying it was "grateful for [Milne's] willingness to put the ABC first in coming to his decision to resign".
In an exclusive interview on Thursday’s 7.30 program, Milne described the week's events as a "firestorm" and said he "wanted to provide a release valve".
"Clearly there is a lot of pressure on the organisation, and as always, my interests have been to look after the interests of the corporation," he said.
Milne has come under scrutiny since firing the then-managing director of the ABC, Michelle Guthrie, on Monday.
The shock moment was immediately followed by public approval of her dismissal by those claiming she had failed to defend the organisation effectively, as well as the formation of an anti-Milne camp, which spurred the release of the email leaks by Fairfax.
Guthrie pasted a copy of the email in a document to the ABC board last week, days before she was fired.
The full version reveals the extent of his concerns about Alberici as chief economics correspondent.
The text read: "After two glasses of red of course there's an agenda. They fricken hate her. She keeps sticking it to them with a clear bias against them. We clear her as ok. We r tarred with her brush. I just think it's simple. Get rid of her. My view is we need to save the corporation not Emma. There is no g'tee they will lose the next election [sic]."
Milne later admitted his concerns about Alberici were partly driven by his desire for the ABC to secure $500 million in funding for Project Jetstream, a major digital transformation project which was being discussed with the government.
Alberici has been under fire since February when the ABC published one of her news articles on the government's corporate tax policies, which contained nine errors.
Malcolm Turnbull later lodged two complaints about the chief economics correspondent, and certain misstatements of fact which were present in her articles.
Milne is known to be a friend and former business partner of former Prime Minister Malcolm Turnbull.
But he said it was not true that Turnbull had rung him in a fury about Alberici's reporting.
The daughter of Italian immigrants who arrived in Australia in 1955, Alberici last week attended a women’s media conference in Italy.
She told ABC Radio Melbourne's Jon Faine that she had not heard that the chairman wanted her sacked until she read it in the news.
"It's disappointing if it's true, obviously. You don't want your chairman to hold such a dim view of you and your journalism," she said.
Alberici said that she had left the commercial media world because she wanted to work for an independent organisation with no outside influence.