Fireworks lit up the sky at the headquarters of Tsai's Democratic Progressive Party (DPP) as thousands gathered to celebrate the historic win over the ruling China-friendly Kuomintang (KMT).
In her first comments to media, Tsai warned that Chinese "suppression" would damage ties with the mainland.
"Our democratic system, national identity and international space must be respected. Any forms of suppression will harm the stability of cross-strait relations," she said.
Her victory came on the same day that outrage erupted over the treatment of 16-year-old Taiwanese K-pop star Chou Tzu-yu, who was forced to record a video apology after angering Chinese netizens by flying a Taiwanese flag in a recent online broadcast.
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Tsai specifically referred to Chou in her address, saying her case had "shaken Taiwanese society".
"This particular incident will serve as a constant reminder to me about the importance of our country's strength and unity to those outside our borders," she said.
In her address to media she pledged to "work towards maintaining peace and stability" in relations with China, but emphasised it must reflect public will.
Jubilant supporters expressed their faith in Tsai as she later addressed the crowds, promising to be a strong leader.
"I'm very confident - we were cheated by Ma's government for so long," said Jimmy Lai, 45.
The United States congratulated Tsai on the victory.
British Foreign Secretary Philip Hammond also congratulated Tsai and said he hoped Taiwan and China would "continue their dialogue to resolve differences and maintain the recent trend of constructive relations".
Tsai remains president-elect until she takes office on May 20.