Morata's stylish volley put Spain ahead in the first half of a Group B clash they dominated with 79.4 per cent of the possession before the break.
But Greece levelled when Anastasios Bakasetas converted from the penalty spot four minutes before the hour mark after Inigo Martínez fouled Giorgos Masouras inside the area.
And although Luis Enrique's side enjoyed the bulk of the chances in the remainder of the game, they were unable to find a winner in an underwhelming start to the campaign.
After a sluggish start, La Roja sparked into life when Dani Olmo rattled the crossbar with a long-range curling effort and, moments later, they were ahead.