The vote allows same-sex couples full legal marriage rights, including in areas such as taxes, insurance and child custody.
Earlier on Friday, Ms Tsai said in a tweet that the island had taken "a big step towards true equality" with the vote.
Same-sex marriage supporters shout during a parliament vote on three draft bills of a same-sex marriage law, outside the Legislative Yuan in Taipei, Taiwan, May 17, 2019. However, adoption rights are still unclear.
The pair has also worked with Taiwan's 20-year-old LGBT movement, which is unusually vibrant for Asia because of free speech protections and lack of a strong organized religion.
Taipei's colorful gay pride parade, one of Asia's largest, puts on display every year the vibrancy of the island's lesbian, gay, bisexual and transgender (LGBT) community.
"I and my partner will go and register our marriage on May 24 when the household registration office opens".
"I congratulate our gay friends for being able to win society's blessing, and I also want to say thanks to those who have different beliefs but still offered support for this law", she wrote in a Facebook post.
Opponents have tabled two other versions which avoid the word marriage, offering something closer to same-sex unions with no adoption rights. "People will find that the world keeps running as normal". However, conservative groups had opposed any changes to the law, staging massive protests against such a move.
The bill's supporters said gay couples had faced discrimination for too long.
"We hope this landmark vote will generate waves across Asia and offer a much-needed boost in the struggle for equality for LGBTI people in the region", said Annie Huang, acting director of Amnesty International Taiwan.
The new ruling will take effect after president Ing-wen passes it into law.
"It's a bit of a regret that this amendment isn't in line with the referendum", he said after Friday's vote. "But Taiwan is a country ruled by law".
Late a year ago, Taiwan voters opposed same-sex marriage in a series of referendums, defining marriage as being between a man and a woman in civil law, though seeking a special law for same-sex unions.
The conservative Coalition for the Happiness of Our Next Generation accused the president on Friday of promoting the same-sex marriage legislation in an effort to pander to young voters and said it could initiate a referendum aimed at overturning the new law. But since then, she has remained largely silent on the issue, preferring to allow the courts and the legislature to take the lead on enacting marriage equality.
Marriages between a Taiwanese citizen and a foreign national would be recognized by authorities if the foreigner's country has allowed legalized gay marriage.