In a statement, the coastguard said one of its vessels had fired warning shots at the Lu Yan Yuan Yu 010 as it headed for international waters.
The coastguard said it had first tried to raise a response by radio. All 32 crew members were rescued, it said.
China has expressed "serious concern" over the sinking.
Chinese foreign ministry spokesman Lu Kang said in a statement that Beijing had "made urgent representations" to Argentina, demanding a thorough investigation.
Illegal fishing is a common problem in the South Atlantic.
"The vessel was hailed over radio (in Spanish and English) and both visual and audio signals were sent to make contact.
However, the vessel turned off its fishing lights and proceeded to flee towards international waters without responding to repeated calls over various frequencies," the coastguard statement said (in Spanish).
"On several occasions, the offending ship performed maneuvers designed to force a collision with the coastguard, putting at risk not only its own crew but coastguard personnel, who were then ordered to shoot parts of the vessel."
China is the world's largest market for seafood and has the biggest long-distance fishing fleet, currently numbering more than 2,000 vessels, reports say.
In 2012 Argentina captured two Chinese vessels it said had been fishing illegally for squid in its exclusive economic zone.
Warning shots were fired.
The latest incident comes despite Argentina's relations with China growing closer in recent years.