The state argues that Connecticut's 2005 civil unions law gives the couples the equality they seek under state law.
The state Supreme Court began hearing the case Monday and was expected to issue a decision later in the year.
Attorneys on both sides say a ruling in the couples' favor could have nationwide implications for states that have adopted or are considering civil union-like legislation.
Currently, only Massachusetts allows same-sex couples to marry. Connecticut, Vermont, California, New Jersey, Maine and Washington have laws allowing either civil unions or domestic partnerships. Hawaii extends certain spousal rights to same-sex couples and cohabiting heterosexual pairs.
Anne Stanback, president of the group Love Makes a Family, and a handful of gay marriage supporters, were among the first to arrive at the Supreme Court on Monday morning. She and her partner of 23 years have not had a civil union because they are waiting for full marriage rights.
"We wanted to make sure we were part of the history," Stanback said.
The hearing also drew gay marriage opponents, including members of the Family Institute of Connecticut.
"We hope the court will realize that something this radical should be left to the people, that something this disruptive, divisive and controversial should be left to the people to decide and not handed down from above," said Peter Wolfgang, the group's director of public policy.
The Connecticut couples, who have been together between 10 and 32 years, say civil unions are inferior to marriage and violate their rights to equal protection and due process.
Married couples have federal rights related to taxes, Social Security beneficiary rules, veterans' benefits and other laws that people in civil unions don't have.
Because civil unions aren't recognized nationwide, other rights, such as the ability to make medical decisions for an incapacitated partner, disappear when couples cross state lines.
The Connecticut couples' claim was dismissed by a lower court last year when a judge said they received the equality they sought when Connecticut passed a same-sex civil unions law. The couples appealed.
The state Department of Public Health and the Madison town clerk's office were named as defendants in the case after denying marriage licenses to the couples based on state Attorney General Richard Blumenthal's advice.
"Our basic argument is, the trial court correctly recognized that there is a rational basis for the state to use a different name for the same rights and benefits accorded same-sex couples," Blumenthal said. "The rights and benefits are identical, whether the union is called a civil union or a marriage."
A bill is pending in Connecticut's legislature to approve same-sex marriage, but leaders of the Judiciary Committee say they want to pull it from consideration this session because they do not believe enough lawmakers would vote to approve it.
Republican Gov. M. Jodi Rell, who signed the civil unions bill into law in 2005, has said she would veto a gay marriage bill. Rell has said she believes marriage is between one man and one woman.
