Neglia was born in Buenos Aires to an Italian Argentine family, in 1929. He took an early interest in the ballet, and at age 12, began taking lessons at the National Conservatory of Music and Scenic Arts, and later from Michel Borowski, a well-known local figure in the ballet. Neglia was accepted into the dance school of the Colón Theatre, the nation's premier opera house, was made part of the opera house's ballet company and was eventually named its primo ballerino.
The nine members of the ballet company boarded a flight at Jorge Newbery Airfield on October 10, 1971, en route to Trelew, a city in Patagonia where they were scheduled to perform. Shortly after takeoff, however, the twin-engine plane stalled, nose-diving into the Río de la Plata and killing all aboard. Their remains lay in state at the Colón, and on the first anniversary of the tragedy, a monument in their honor was unveiled on Lavalle Plaza, near the opera house.