The law makes it easier for unmarried co-habiting couples to co-own property and make medical decisions.
It applies to both heterosexual and same-sex couples.
"From today, Chile is different. Chile is for everyone," Roxana Ortiz said after formalising her relationship with Virginia Gomez in front of a judge.
"It was all very emotional. Our families were here, everyone was shedding tears," she told reporters.
Government spokesman Marcelo Diaz said the law "was unthinkable a few years ago" in Chile, one of the most socially conservative countries in the region.
Gay rights advocates say they hope it is a first step on the road towards allowing same-sex marriages.
Co-habiting couples who register under the new law will be able to claim pension benefits and inherit property if their civil partner dies.
It will also make it easier for people to get custody of their partner's child if he or she dies.
The Law for All Families was passed in January after years of lobbying by activists.
Officials said 1,600 couples had already signed up to register their unions.
The government says as many as two million people are currently co-habiting and could now have their unions legally recognised.
Chile only legalised divorce in 2004 and abortion remains totally banned.
Many countries in the region allow civil unions but gay marriage has been legalised nationwide only in Argentina and Uruguay, while it is also allowed in some Mexican states.
In Brazil, Congress has not legalised gay marriage but a key judicial body has instructed registry offices they must accept same-sex marriages.
Dozens of same-sex couples in Chile have been celebrating as a new law recognising their unions came into effect.