From a 1998 Airbus publication:
ETOPS milestones
Airbus operators have been operating their A300 twinjet aircraft across the North Atlantic, the Bay of Bengal and the Indian Ocean under the 90-minute ICAO rule since 1976. However, ETOPS officially began in 1985 with the newly issued ETOPS criteria.
In 1985, the first ETOPS operations (90 minutes) were made in February by TWA with a 767 and in June by Singapore Airlines with an A310.
In April 1986, Pan Am was the first to inaugurate transatlantic revenue service with A310-200 and A310-300 aircraft. In less than five years, more than 20 operators joined the two pioneers in Airbus ETOPS operations.
http://www.737ng.co.uk/AIRBUS%20ETOPS%20Guide.pdf
I would say the 767, not the 777, was the answer to the A300.