Tsai and her Democratic Progressive Party (DPP) won a landslide victory on Saturday as voters turned their backs on closer China ties.
The DPP has traditionally backed independence for the island, but Tsai has moderated its rhetoric, promising to maintain the "status quo".
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Beijing "will not hold unrealistic delusions" about her, Zhou added, saying that whether cross-Strait relations take "the road of peace or antagonism, it's up to Tsai Ing-wen to make the choice".
Although Taiwan is self-ruling after it split with China following a civil war in 1949, it has never declared independence and Beijing still sees it as part of its territory awaiting reunification.
Taiwanese support for Tsai and the DPP surged after voters became increasingly uneasy about a rapprochement with China under outgoing President Ma Ying-jeou of the Kuomintang.
Tsai wasted no time in warning China that "suppression" would harm cross-strait ties in her first comments to international media following her win, adding that "our democratic system, national identity and international space must be respected".
An editorial in Monday's English-language China Daily newspaper insisted that the Kuomintang lost the election due to issues such as rising unemployment and inequality, rather than its Beijing-friendly approach.
But it added that Tsai's policy towards the mainland "remains ambiguous".
"She has a responsibility to keep the peaceful development of cross-Straits relations on track," it said.
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