The new act means the Isle of Man is the only place where both gay and straight couples can enter into civil partnerships.
The young couple, who live in Douglas, said they were "very proud".
The pair entered into their partnership in front of two witnesses at a small ceremony in the island`s capital.
Couples who register a civil partnership on the island acquire the legal status of civil partners and gain similar rights and responsibilities to those of a married couple.
Ms Cosson, an au pair from France, told the Guardian newspaper that she and Mr Hodgson "do want to get married one day" but a civil partnership gives them rights under the law.
"It helps couples move forward without having to get married right now. In France, everybody can get a civil partnership."
Matt Hawkins from the Equal Civil Partnerships Campaign said they were "delighted".
"It`s an opportunity we want to see extended to all different-sex couples living in Britain and we hope that Westminster will soon follow the lead of the Isle of Man in making this a possibility.
For us, it is simply about extending choice and giving everyone the right to have their relationship recognised and celebrated in a way that works for them."
Civil partnerships were introduced on the Isle of Man for same-sex couples in 2011 - seven years after the UK.
UK case dismissed
Next month, London couple Charles Keidan and Rebecca Steinfeld will go to the Court of Appeal to argue that denying UK same-sex couples civil partnerships breaches their human rights.
In 2014, they were told they could not have a civil partnership because they did not meet the legal requirement of being of the same sex.
They took their case to the High Court in January, but their claim was dismissed.
Ms Steinfeld tweeted today: "BIG NEWS: 1st ever different-sex civil partnership in Britain has just taken place on the Isle of Man! Hooray!"
Same-sex marriages were given the go ahead in the Isle of Man in July following the granting of the Marriage and Civil Partnership Act being granted Royal Assent.
Any two persons, regardless of sex, can register a civil partnership on the Isle of Man provided they are not legally married or in a civil partnership.
The minimum age to register a civil partnership on the island is 16.
In August, Luke Carine and Zac Tomlinson became the island`s first same-sex couple to marry.
Marc and Alan Steffan-Cowell became the first Manx couple to convert their civil partnership to marriage a week earlier after choosing not to have a ceremony.
Homosexuality was still illegal under Manx law until 1992.
