The rover, which landed on the red planet in Aug 2012, used a drill carried at the end of its robotic arm to bore into the rock and collect a sample from its interior.
The rock is believed to hold evidence about long-gone wet environments.
"The most advanced planetary robot ever designed is now a fully operating analytical laboratory on Mars," said John Grunsfeld, NASA associate administrator for the agency's Science Mission Directorate.
"This is the biggest milestone accomplishment for the Curiosity team since the sky-crane landing last August."
Curiosity will soon use its laboratory equipment to analyse rock powder collected by the drill.
Small portions of the sample will be processed by the rover's Chemistry and Mineralogy (CheMin) instrument and the Sample Analysis at Mars (SAM) instrument.
Curiosity landed safely in the Gale Crater for a two-year mission to determine if life exists now or has in the past, to characterise the climate and geology, and prepare for future human exploration on Mars.
The probe is equipped with 10 science instruments weighting a total of 75 kg for detailed geological, geochemical, atmospheric and climate studies.
It also has instruments to detect possible traces of water and organic compounds.
Meanwhile, NASA said that it had started environmental testing for another Martian mission, Maven (Mars Atmosphere and Volatile EvolutioN).
The aim of the mission, scheduled for launch in Nov 2013, is to study the Martian upper atmosphere.
During the environmental testing phase, to be held in the Lockheed Martin Space Systems facilities near Denver, the orbiter will undergo a variety of tests that simulate the extreme temperatures, vacuum and vibration it would experience during the course of the mission.
In Aug, the spacecraft is scheduled to be taken to NASA's Kennedy Space Center, where it will undergo final preparations for launch.
Maven, approved by NASA in Oct 2010, is a robotic exploration mission to understand the role that loss of atmospheric gas to space played in changing the Martian climate through time.
It will investigate how much of the Martian atmosphere has been lost over time.