Shitaye Sissay scored the lone goal on 14 minutes into the first half but the Ethiopians failed to take advantage of their numerical strength to rattle in more goals.
Defender Gladys Enti was shown the exit in the 85th minute before midfielder Safia Rahman was marched off three minutes later. The return leg will be played in Accra in a fortnight.
In other matches, Cameroon were held goalless at home by Equatorial Guinea and South Africa defeated Tunisia 1-0. African champions Nigeria face Namibia in the last qualifier.
The eight teams left are fighting for two places to represent the continent at the Games.
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