Vidal scored from the penalty spot in the 67th minute after being pulled down by Miller Bolanos before Vargas latched on to a pass from Alexis Sanchez to double Chile's lead with a low right-foot shot six minutes from time.
Ecuador's best chance came from an Enner Valencia header that bounced back off the crossbar with the score still at 1-0.
Despite their victory, the Chileans created very little for much of the tie and will need to improve if they are realize their hope of a first Copa America title at the 37th attempt.
They face Mexico in their next match on Monday.
The Mexicans take on Bolivia on Friday in their Group A opener before heavyweights Argentina, Brazil and Uruguay, the defending champions, get their campaigns underway over their weekend.
Sanchez could have had two goals inside the first four minutes at Santiago's national stadium but scuffed his first effort and then failed to beat goalkeeper Alexander Dominguez with a lob.
With the match goalless at halftime, Chile coach Jorge Sampaoli threw on Napoli striker Vargas in a bid to break the deadlock.
The change seemed to breathe life into the Chilean front line and they gradually took control as the game opened up before running out deserved winners.
Chile's win extended their fine run against the Ecuadorians at the Copa America.
In 14 encounters dating back to 1939, Chile have won 12 and found the back of the net on all 14 occasions. They have no better record against any other side in the tournament.
Ecuador have now won only one of their last 16 Copa matches in a dismal run dating back to 1999. They face Bolivia next. – Reuters
Original source: Vidal gets Chile off to winning start