Taiwan’s top court ruled that current law defining marriage as exclusively between a man and a woman was unconstitutional, paving the way for the first same-sex marriage law in Asia.

Both celebrations and protests broke out across Taiwan on Wednesday after the island’s constitutional court ordered the legislature to change the law to permit gay civil unions within two years. If the legislature failed to act, the court said same-sex couples could register to marry in any case.

“Today is a watershed,” said Chen Yi Chen, 37, a lawyer in Taipei who said she and her girlfriend of six years had until recently not seriously talked of marriage because the prospect seemed so unlikely.

The plaintiff in Wednesday’s case, Chi Chia-Wei, had pushed for more than three decades to advance same-sex marriage rights. The effort gained pace in recent years as Taiwan emerged as a beacon for gay-rights supporters in Asia, annually hosting a gay rights parade that attracts tens of thousands of participants, some from around the region. President Tsai Ing-wen, who was elected last year, has expressed support for same-sex marriage.

By early Wednesday evening, the news was the eighth-highest trending topic on China’s Twitter -like Weibo, with users posting rainbow flags and hearts and at least one hashtag racking up more than 16 million views.

The court said the need to create “permanent unions of intimate and exclusive nature are equally essential to homosexuals and heterosexuals,” citing freedom of marriage as important to safeguard human dignity.

Taiwan’s greater tolerance for sexual minorities is unusual within Asia, particularly in countries with larger Christian populations and amid a rise of a more conservative Islam in Indonesia and other areas. Two men were publicly caned in Indonesia on Tuesday as punishment for having same-sex relations.

Neither same-sex marriage nor civil unions are recognized in mainland China.

Still, the issue has proved polarizing even in Taiwan, with both supporters and opponents of same-sex marriage—including Taiwan’s small, but vocal Christian population—staging large rallies last year to support their positions.

In the run-up to the verdict, opponents staged protests and circulated commentary online, including in the form of satirical cartoons, one which depicted a large building labeled “Marriage” cracking after a grey column was removed and replaced with one painted in rainbow stripes.

The self-ruled, democratically governed island split from mainland China after the civil war with the Communists ended in 1949. Beijing has increasingly worked to sideline Taiwan, especially following the election of Ms. Tsai, whose Democratic People’s Party officially supports independence. This week, under pressure from Beijing, Taiwan was denied observer status at the World Health Assembly in Geneva.

Advocates for gay marriage in Taiwan say they hope the move can help inspire other nations in Asia.

“We have a very tricky diplomatic position,” said Wayne Lin, chair of the Taiwan Tongzhi (LGBT) Hotline Association, a support group.

“But I think Taiwan can show that we are a country who cares about human rights—and we can show that LGBT rights isn’t something Western, it’s something for all Asian countries.”

On her official Facebook page, Ms. Tsai appealed for tolerance and understanding in the wake of the verdict. “No matter what your stance is on same-sex marriage, this is the moment when we should view everyone around us as brothers and sisters,” she wrote.

—Jenny W. Hsu contributed to this article.