Uruguay, World Cup semi-finalists two years ago, fell behind to an eight-minute penalty converted by burly striker Felipe Caicedo after he had been brought down by central defender Diego Lugano.
Cavani secured a point for Uruguay when the ball was poorly cleared from an attack down the left and ran loose to the striker on the right, from where he shot left-footed low into the bottom of the far corner.
"Their goal woke us up, we began to put more intensity into our game and it was like that right to the end, that's why we managed to draw," Cavani told reporters.
"I think they had the luck to find [an equaliser] when there were two rebounds as we tried twice to clear [the ball]," Ecuador's Colombian coach Reinaldo Rueda said.
Ecuador, who were in control for long spells and especially in the first half, were stunned when they were denied a penalty by Paraguayan referee Carlos Amarilla just past the hour when they were still a goal ahead.
Uruguay goalkeeper Fernando Muslera appeared to trip Cristian Benitez with his outstretched leg as the striker tried to run past him in a solo breakaway but instead Amarilla booked the Ecuadorean for diving.
Striker Luis Suarez might have equalised in the 43rd minute but his shot hit the bar and he was denied a winner two minutes from time when his header was saved by Alexander Dominguez.
Ecuador winger Antonio Valencia was sent off in the fourth minute of added time after a second booking for obstruction.
