Ethiopia took a shock lead in the 6th minute thanks to Birkie Amare, who slotted home from a goal mouth scramble.
In the 52nd minute the Falcons equalised through captain Rasheedat Ajibade, who volleyed home a long long range shot from out the box.
The Falcons thought they had grabbed the winner in the 92nd minute as substitute Ifeoma Onumonu headed home but it was ruled out for offside.
The second leg will come up on Tuesday, October 31 in Abuja.
After making a first appearance at the Olympics in Sydney 2000 and going on to feature at Athens 2004 and Beijing 2008, the Falcons have failed to qualify for the last three editions.
They failed to qualify for London 2012, Rio 2016 and Tokyo 2020.
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