Five Supreme Court justices unanimously allowed an appeal by Rebecca Steinfeld, 37, and Charles Keidan, 41.
They are currently prevented from having a legal union through the route of civil partnership because the Civil Partnership Act 2004 says only same-sex couples are eligible.
The academics, who live in Hammersmith, west London, suffered defeat at the Court of Appeal in February last year, but were given the go-ahead in August for a Supreme Court hearing.
The panel of Supreme Court justices, including the court's president, Lady Hale, heard the couple's case in May and announced their decision on Wednesday.
The judges granted a declaration that the 2004 Act was "incompatible" with human rights laws on discrimination and right to a private and family life.
Lord Kerr, announcing the court's decision, said the Government "does not seek to justify the difference in treatment between same-sex and different sex couples".
He added:"To the contrary, it accepts that the difference cannot be justified."
What the Government sought was "tolerance of the discrimination while it sorts out how to deal with it".
He concluded: "That cannot be characterised as a legitimate aim."
Lord Kerr said it was "salutary to recall that a declaration of incompatibility does not oblige the Government or Parliament to do anything".
The couple, who have two daughters aged nine months and two, claimed the Government's position is "incompatible with equality law".
During the hearing, their barrister, Karon Monaghan QC, told the court they have "deep-rooted and genuine ideological objections to marriage" and are "not alone" in their views.
She said matrimony was "historically heteronormative and patriarchal" and the couple's objections were "not frivolous".
Ms Monaghan added: "These are important issues, no small matters, and they are serious for my clients because they cannot marry conformable with their conscience and that should weigh very heavily indeed."
The Court of Appeal agreed that the couple had established a potential violation of Article 14 of the European Convention on Human Rights, which relates to discrimination, taken with Article 8, which refers to respect for private and family life.
But, by a majority of two to one, the judges said the interference was justified by the Government's policy of "wait and evaluate".
The Government said it was decided, after public consultations and debate in Parliament, not to extend civil partnerships to opposite-sex couples, abolish them or phase them out at that stage.
The aim was to see how extending marriage to same-sex couples impacted on civil partnerships before making a final decision which, if reversed in a few years' time, would be disruptive, unnecessary and extremely expensive.
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