Whether the rapid warming of the Arctic, particularly the Barents-Kara Sea (BKS), substantially affects the Eurasian winter climate has been debated for over a decade. Here, we use an extended dynamical adjustment method to separate the effects of internal dynamics and thermodynamically forced BKS warming on atmospheric circulation, relying solely on observations. Evidence shows that the observed link between BKS warming and Eurasian cooling is influenced by both atmospheric internal variability and forced BKS warming. Internal variability, particularly the Arctic Oscillation, predominantly contributed to the observed Eurasian cooling from 1991 to 2012. While BKS warming has a weaker impact on Eurasian cooling on interannual to interdecadal timescales, it notably affects multidecadal scales, contributing to the observed "warming hole" in central Eurasia during 1980-2022. Our findings suggest a weak but non-negligible Eurasian cooling response to BKS warming on multidecadal timescales. These findings advance the understanding of the complex causal relationships between Arctic and mid-latitude climates.