Nicolas Lodeiro had given Oscar Tabarez's men a sixth-minute lead before Jose Quintero was sent off for Ecuador.
That made an already difficult task even tougher and Uruguay capitalised on their numerical advantage in the first half through goals from star duo Edinson Cavani and Luis Suarez.
Uruguay comfortably saw out its win, added to by an Arturo Mina own goal in the second half, to secure a dream start in Group C, which also includes Japan and two-time reigning champion Chile.
Diego Godin almost gifted Ecuador a first-minute opener after an underhit back pass before Uruguay opened the scoring.
Suarez's cross from the right found Lodeiro, who took two beautiful touches before unleashing into the bottom corner on the half-volley.
Uruguay was already in control and the game turned further in its favour in the 24th minute.
Quintero made contact with the face of Lodeiro in an aerial challenge and was shown a yellow card, only to be sent off by the referee after a lengthy VAR review.
Uruguay pushed on from there, Cavani forcing two fine saves from Ecuador goalkeeper Alexander Dominguez, including one onto the post after a delightful flick.
But the Paris Saint-Germain forward would double his side's lead in the 33rd minute with his first Copa America goal, acrobatically volleying in from close range after Godin centred a header.
Uruguay went further ahead just before the break, Suarez converting at the back post after Martin Caceres flicked on a corner from Lodeiro.
In control with its three-goal lead, Uruguay was content to simply dictate possession to begin the second half.
But it sealed its win in the 78th minute, Mina scoring an own goal that was initially ruled out for offside before being overturned by the VAR.