The spread of Buddhism from China into Korea and Viet Nam was a major cultural influence emanating from China and affecting those two other cultures. A very clear and concise discussion of the question you raised can be found here:
The final paragraph provides a succinct summary:
Chinese culture spread to the sedentary agricultural populations of Korea, Japan, and Vietnam in the first millennium C.E. Chinese writing, bureaucratic organization, religion, and art all made impressions on the indigenous cultures. In general, the local elites of the three regions actively sought to emulate Chinese models. Differences within the three areas resulted in divergent outcomes and alternative mixes of the indigenous and the imported. China was able to establish direct control over Korea. In Vietnam, Chinese influences mingled with Indian cultural contributions. Only Japan remained permanently independent of China and, thus, was able to selectively adapt Chinese models to Japanese needs. In East Asia, as a whole, cultural exchanges took place in isolation from the rest of the civilized world.