Known as Nico, the robot can identify almost exactly where its arm is in space based on the mirror image.
Yale University's Justin Hart and Brian Scassellati have taught Nico to recognise its arm's location and orientation down to accuracy of two centimetres in any dimension.
This is a feat of spatial reasoning that no robot has ever accomplished before, the 'New Scientist' reported.
Nico is at the centrepiece of a unique experiment to see whether a robot can tackle a classic test of self-awareness called the mirror test.
To pass the test, an animal has to recognise that a mark on the body it sees in the mirror is in fact on its own body.
Only dolphins, orcas, elephants, magpies, humans and a few other apes have passed the test so far.
Precise recognition of where its body is in space will be key if Nico is to get to grips with the mirror test, which by its nature is performed in 3D.
"What excites me is that the robot has learned a model of itself, and is using it to interpret information from the mirror," said Hart.
Mary-Anne Williams of the University of Technology Sydney, Australia, points out that robotic self-awareness is crucial if robots are ever going to work safely alongside humans.
The work was presented last month at the Conference on Artificial Intelligence in Toronto, Canada.