The movie examines the illegal abortions that were done in communist-era Romania when abortions were prohibited -- a topic frequently used as propaganda by abortion advocates to argue for legal abortions.
Featuring Anamaria Marinca, it focuses on the "horrors" a student endures to help her friend have an abortion. It's named after the age of the baby killed in the illegal abortion.
"Thank you very much for this award, it honors me greatly," director Cristian Mungiu said after receiving it.
"[The] BBC is for me a symbol of quality, professionalism, tradition and equilibrium, therefore, I can not think of a better recommendation for those who haven't yet seen 4 Months 3 Weeks, 2 Days, to go and watch it, than a BBC award,” he added.
Actors John Hurt and Adrian Lester, director Asif Kapadia and London film festival director Sandra Hebron served on the BBC judging panel this year.
The director spent very little on the low-budget film and almost didn't have enough money to make it.
In an apparent attempt to make the movie less polemic Mungiu does include a shocking image of the aborted baby and a graphic description from the abortion practitioner of the abortion procedure involved.
One review of the movie said it also focused on the abortion practitioner's "chilling exploitation of the women's dilemma."
"Because of the pressure of the regime, women and families were so much concerned about not being caught for making an illegal abortion that they didn't give one minute of thought about the moral issue," Mungiu told reporters about his film after winning at Cannes.
"It was either you or them getting you for what you did."
He said he put the image of the dead baby after the abortion on the screen to making a point -- "people should be aware of the consequences of their decisions."