"The election result doesn't mean the Taiwanese people want to abandon sovereignty, nor does it mean the Taiwanese people will give up Taiwan's autonomy", Tsai said in a New Year's Day address.
His term is to end on May 19 next year.
Beijing has increased its military presence around Taiwan since President Tsai was elected in 2016, in fear that she is seeking formal independence despite her claims that she is comfortable sticking to the status quo.
"It's important because the global community, and [mainland] China, will be watching", J. Michael Cole, a Taipei-based expert with the University of Nottingham's China Policy Institute, said.
Tsai resigned the party chairmanship but stayed on as president, staying above the fray in the vote to replace her.
You, who served as the deputy head of the Mainland Affairs Council and secretary-general of the Straits Exchange Foundation during former president Chen Shui-bian's (陳水扁) time in office from 2000 to 2008, was mainly backed by a group opposed to Tsai.
The by-election was held to replace President Tsai Ing-wen (蔡英文), who resigned as DPP chairwoman November 24, 2018 in the wake of her party's heavy losses in the local government elections earlier in the day. "If the worldwide community does not make Taiwan a democracy under Chinese threat, then we would like to ask, which country is next", said Tsai. "Any major departure from long-standing policy under President Tsai could alarm worldwide partners and give Beijing ammunition to further crack down on Taiwan". "Party members voted for continuity", he told Agence France-Presse (AFP), adding that other countries "will also be reassured". He beat polling specialist You Ying-lung, who has been critical of the president.
Since the founding of the people's Republic of China in 1949, Beijing considers the island Republic of Taiwan as its own territory, and threatened again and again with a re-conquest.
Yen Chien-Fa, a political analyst at Chien Hsin University, said whoever comes out on top will have significant influence on the 2020 campaign and whether Tsai is the candidate. A DPP swing towards its more radical wing might also worry Washington.
The United States diplomatically recognises China over Taiwan but it remains Taipei's staunchest political and military ally.