From what I remember from Biology, a muscle has to relax for another to constrict. The muscles involved are in the wrist and the rest of the hand. The actual grasping between finger and thumbs is unique to humans, no other primate can do this.
most primates have opposable thumbs. fine work between finger and thumb is unique to humans but gorillas, chimps orang-utans and old world monkeys can bring thumb and forefinger together in a gross grasping motion. Japanese macaques are particulary good at it. However all these require the grasped object to be quite large. They couldnt pick up a pin, for example.