Nicolas Lodeiro opened the scoring after just six minutes before Ecuador full-back Jose Quintero was dismissed for a flailing arm.
With a numerical advantage, Uruguay ran riot at the Mineirao stadium in Belo Horizonte as Cavani and Suarez netted before the break. The rout was completed 12 minutes from time with an Arturo Mina own goal.
"We had chances in the first half, we managed to take them and that gave us the tranquility to manage the game," said Cavani, who finally scored his first Copa America goal in his fourth participation.
"It was an important step."
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It was the ideal beginning for Uruguay in a tough Group C alongside champions Chile and guests Japan, who meet on Monday in Sao Paulo.
"We're not as bad as that," lamented Ecuador's Colombian coach Hernan Dario Gomez.
"It was a match that got away from us, especially in the first half when it got away from us in every aspect."
Uruguay were off to a dream start.
Suarez crossed from the right and Lodeiro proved the coolest customer in the Mineirao stadium as he beat one defender with his chest control, flicked the ball over another and then hit a crisp, low volley into the bottom far corner.
"We knew they would be very tough opponents. The important thing was start the game with maximum concentration and we managed that," said Cavani.
Uruguay were rampant and had the ball in the net soon afterwards only for Cavani to be given offside in the build-up.
Ecuador brought a physicality to the contest that upset the Uruguayan players and Quintero was perhaps lucky to escape punishment when catching Diego Laxalt with an elbow in an aerial challenge.
But moments later, his arm made contact with Lodeiro's face when challenging for another header, this time drawing blood.
Brazilian referee Anderson Daronco -- whose bulging arm muscles put even Ecuador's strapping players to shame -- initially showed the full-back a yellow card but after consulting VAR he changed that to red.
That left Ecuador with a mountain to climb.
On 27 minutes, Alexander Dominguez made a brilliant low save to turn behind Cavani's half-volley from 15 yards.
Paris Saint-Germain forward Cavani was denied again when Dominguez stunningly tipped his clever back-flick onto the post one-handed.
But it was third time lucky on 33 minutes as Diego Godin headed the ball into the danger area and Cavani acrobatically rasped a bicycle kick from seven yards past a helpless Dominguez, his 47th international goal.
Nahitan Nandez wasted a good chance, shooting straight at Dominguez but Suarez put the game to bed a minute before the break, stealing in at the back post to poke home a flick-on from Martin Caceres.
With the game all but over as a contest, the second half was a damp squib of an affair.
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Uruguay had the ball in the net soon after the restart but Cavani was offside again.
Ecuador showed plenty of commitment, continuing to rough up Uruguay's more technical players, but never looked like they believed they would get anything out of the game.
Uruguay looked content to play a patient possession game without pushing for more goals.
Cavani did send Suarez scampering in on goal after a defensive mistake but Dominguez was quickly off his line to snatch the ball off the Barcelona forward's toes.
The fourth came in calamitous fashion as Gaston Pereiro tried to pick out Suarez from Cavani's cross even though he was only six yards from goal.
His header dropped to Ecuador center-half Mina, who inadvertently volleyed into his own goal.
"The important thing was that the team won and now we'll prepare for the next match," against Japan on Thursday, said Suarez, who notched his 57th goal for Uruguay.
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