Bangladeshi Mahfuza Akhter Kiron overcame Australia's prominent women's football campaigner Moya Dodd 27-17 in Monday's vote to secure the win.
Asked by BBC World Service to name the current Women's World Cup holders, Kiron named USA at the third attempt.
"This is very disappointing," said two-time World Cup winner Carli Lloyd.
Kiron answered "Korea" and "Japan" before settling on USA.
Lyon forward Alex Morgan joined American compatriot Lloyd in being critical on social media.
Fifa's council is made up of six confederations - representing Asia, Europe, Africa, North and Central America, South America and Oceania - and each must have one female representative.
Dodd was expected to sit on the Asian confederation, which voted at its 27th Congress in Bahrain.
The former Australia international - now a practising lawyer - had even served on world football's executive body as an appointed member between 2013 and 2016 and was a leading voice promoting women's football.
"Naturally it is disappointing," said Dodd, 52. "I'd hoped I had done enough in the few years I was part of Fifa to persuade people that I should have another shot at it."
Kiron told the BBC the women's game in Asia does not have enough sponsors and she hopes to deliver more coaches and education.
"This is like a dream come true," said Kiron. "I want to do something for Asian women's football. I am a sports organiser. For the last 10 years I worked with football and business."
Kiron's victory came through a head-to-head vote when other candidates withdrew.
Fifa's newly elected female representative for Asian football has been criticised for struggling to name the current women's world champions.