Mr Corbyn was elected leader last September after party members and registered supporters voted for him in huge numbers.
Eagle declared her intention to run for leader over the weekend, saying she was the candidate who could "heal the party".
She reinforced her belief that Jeremy Corbyn is unable to provide the leadership necessary in the aftermath of the European Union referendum, and said that she "will not stand back and allow Britain to become a one-party Tory state".
Mr Corbyn said it would be "irresponsible" for him to quit because of the mandate he had been given by members, and insisted there had been no "wobble" over his leadership.
The ex-minister said she had no choice but to move against the Labour leader because Britain was in danger of becoming a "one-party Tory state" under his leadership.
The constituency Labour party has said it supports Mr Corbyn.
"I think she's the Empire Strikes Back candidate - she voted for Iraq, she voted for tuition fees".
Following a meeting at Labour headquarters in central London lasting more than five hours, the NEC voted by 18 to 14 in his favour.
Angela Eagle, who at the weekend announced her leadership campaign, said that she welcomed the contest ahead and was "determined to win it".
The 55-year-old said her background as a "good, sensible, down-to-earth woman with northern roots" would help her lead the party.
"I've tried to find a way forward for the party between two apparently irreconcilable decisions".
Though many party members continue to stand behind the veteran politician, Corbyn has faced several clashes with Labour MPs in the time since taking over party leadership, resulting in a no-confidence motion in his leadership.
Angela Eagle, a senior gay Labour MP, has officially launched her bid for leadership.
Mr. Corbyn and his supporters have argued that, as the incumbent, he should automatically be put on the ballot.
"I don't go in for suicide missions", she said.
It has been suggested that Corbyn would need to find the support of 51 Labour MPs in order to be on the ballot, as Neil Kinnock did in 1988 when he was challenged by Tony Benn.
Mr Corbyn released a statement criticising the attack and said...
But nearly simultaneously, the party's election chief, the MP Jon Trickett, urged his colleagues to begin preparations for a general election, as it became clear that Home Secretary Theresa May is likely to become prime minister within days. "I am my own woman".