The 26-15 vote by the state House of Representatives was heralded by gay rights groups. The state Senate passed the bill last week on a 13-6 vote.
"Parties who enter into a lawful civil union in Delaware, or whose legal union is recognized as a civil union under Delaware law, will have all of the same rights, benefits, protections and responsibilities as married persons under Delaware law," the measure says.
"Today, we celebrate a victory for all Delaware families who will have the tools to protect themselves in good times and in bad," said Joe Solomonese, Human Rights Campaign president, in a statement.
Delaware Gov. Jack Markell has said he will sign the bill, according to the Delaware News Journal.
"The fact that the Delaware General Assembly passed civil union legislation on the first attempt is a great testament to how far public opinion has come on equality issues," said Lisa Goodman, president of Equality Delaware.
The bill does not allow same-sex couples to obtain marriage licenses, and religious institutions retain the right to refuse to solemnize such unions. After it is signed, the bill will take effect January 1.
The new law will make Delaware the eighth state to offer civil unions or comprehensive domestic partnerships for same-sex couples, according to the Human Rights Campaign. The others are California, Nevada, New Jersey, Oregon and Washington. Couples in Illinois can begin applying for civil union licenses on June 1, according to the Human Rights Campaign, and Hawaii's law takes effect on January 1.