Dogs do the same to bond with other dogs, scientists report in the journal, Royal Society Open Science.
They think dogs may be showing a basic built-in form of empathy, enabling them to pick up on emotions.
And the phenomenon may have emerged in our canine companions during the process of domestication, say scientists from the Natural History Museum, University of Pisa.
Split-second mirroring
Until now, the idea - involved in social bonding - has only been described in humans and non-human primates such as chimps and orangutans.
It`s why humans automatically mirror a smile or a laugh, enabling the sharing of emotions.
"We demonstrated that rapid mimicry is present in dogs and it is an involuntary, automatic and split-second mirroring of other dogs," lead researcher Dr Elisabetta Palagi told BBC News.
She said it appears that dogs are showing a basic form of empathy where they are able to instantly pick up on the emotions of other dogs through their facial expressions and body movements.
This is regarded as the first step towards more complex forms of empathy.
"A dog while playing with another dog can read their motivation and the emotional state of the other dog by mimicking the same expression and body movement of the other dog," Dr Palagi explained.
"This phenomenon is present also in humans and in other primate species."
Rapid mimicry
The team of three researchers - working with the Unit of Cognitive Primatology and Primate Center in Rome - videotaped dogs playing in a park in Palermo, Italy.
They analysed the way the dogs were interacting, including signals used to show when a dog was being playful, such as:
Crouching or "bowing" down on its front legs
Relaxing its mouth to reveal some of its teeth.
After analysing 50 hours of video, they found that dogs were able to mimic the facial expressions and movements of other dogs in a split-second.
This "rapid mimicry" is an automatic and involuntary response, rather than the result of training, they say.
Human-dog relationship
But Dr John Bradshaw of the University of Bristol School of Veterinary Science said more research was required to establish if dogs are really able to sense what emotions are uppermost in the minds of other dogs.
"Domestic dogs are exquisite readers of body-language, both that of other dogs, and, uniquely, our own - which is why they`re so easy to train," he said.
"They also love to play, so quickly learn that imitating the actions of their play-partner means that the game goes on for longer.
"But science has yet to show that dogs have any understanding of other dogs` thought-processes, or emotions."
Dogs are known to be able to respond to human emotions, such as copying a yawn, suggesting they show some basic aspects of empathy.
This capacity may have evolved in dogs as they were domesticated or could have been present in the wild ancestors of canines.
The Italian researchers plan to study wolves to answer this question and shed more light on the complex relationship between humans and dogs.
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