Pegasus came first. Pegasus was a creature from Greek mythology who was the child of Medusa and Poseidon. When Medusa was still a pretty girl she was raped by Poseidon and became pregnant, but soon after she was transformed into the snake-haired Gorgon we all know her by. Apparently monster women can't give birth so Pegasus was stuck inside her, unable to be born. When Perseus beheaded her, Pegasus sprang from her blood from her neck. His name comes from the Greek "pegai" meaning "springs of the Ocean" and "sus" meaning "bridled horse." He was featured heavily in other myths, especially the story of Bellephron who was given a magic girdle by the gods so that he may ride Pegasus and together they killed the Chimera. When Bellephron tried to fly Pegasus to the top of Mt. Olympus to see where the gods lived, the magic wore off and Pegasus bucked him off so that he fell to his death. In art it was often associated with the 9 Muses and a symbol for creative inspiration.
A unicorn does appear in ancient Greece but not in mythology. Rather ancient greek naturalists included descriptions of unicorns in natural histories, the earliest being Ctesias, who was from 5th century BC. This is 300 years after the first stories of Pegasus which appeared around approximately 8th century BC. These ancient Greek descriptions vary significantly from our own current notions of unicorn and were not uniform among authors. All of them do agree that it is an animal that exists in India. Our image of a unicorn started in medieval times. The medieval uniocorn is not simply a white horse with a horn, however, instead it has a billy-goat beard, lion's tail and cloven hooves. Some traits associated with traditional unicorns:
horn could neutralize poison and had aphrodisiac qualities
could only be tamed by a virgin, otherwise it was a fearsome beast, unable to be captured
it was a metaphor for purity and chaste love
mortal enemy of elephants. would use horn to pierce an elephant's belly if their paths crossed
A pegasus with a unicorn name actually has a name. It is referred to as an Pegacorn. But as it is a more recent development, it has no standardized mythology behind it.