The 36-year-old Portuguese is leaving her role with the Iran women's side to join Clermont Foot from next season, the Ligue 2 side confirmed on Wednesday morning.
A well-respected tactician, Costa has largely managed in the women's game but has history in the men's too, having coached at age-group level for Benfica and scouted for Celtic from 2008-2011.
A Celtic spokesman said: "She did an excellent job for us and we know she will give her new position the same high level of commitment and dedication which she gave to Celtic."
Clermont striker Remy Dugimont told L'Equipe: "We learned together this morning. We were told the new coach was this lady. It is a surprise, a big surprise, because a lot of names were circulated; all coaches who were more or less unemployed.
"It came out of nowhere and now we have to get to know. It is good, it will create a buzz and will be a unique experience."
Dugimont does not believe Costa's relative lack of experience in the men's game will be a problem, nor does he see issues arising with respect.
"I'm not macho. She has worked in the profession and all the boys know how it works," he added.
"At first it will be a little weird but it will only last a few days. We don't not know how it will happen but if the president has chosen to bring her, we hope it will be okay and I think there are no problems."
Dugimont's view is shared by team-mate Anthony Lippini.
"I spoke with the physio and we spoke about the first time a woman entered the army... A very macho environment," he said.
"Now there are many women in the army and it may be the same in football. I cannot wait for next season to discover. I'm really curious."
The news was also welcomed by Najat Belkacem, France's Women's Minister, who said on Twitter: "Bravo Clermont Foot for understanding that giving a place to women is the future of professional football."
Costa, who has also coached the Qatar women's side, will join Clermont officially at the end of the season which has two games to run. She will replace Regis Brouard.
"It should help the club enter a new era," the club's statement said. Clermont currently sit 14th in the 20-team Ligue 2.
Prior to Costa, Carolina Morace coached Italian Serie C1 side Viterbese for two games 15 years ago. She also won the Regional Lisbon League with male side Cheleirense in 2006.
Costa has a good track record in the women's game, having won two Portuguese titles with Sociedade Uniao 1 Dezembro and winning promotion with Odivelas.
Clermont plan to announce more details of Costa's deal when she is presented at the end of the season.