The win fired the holders into the quarter-finals of the competition.
Chile took the lead in the eighth minute when Jose Fuenzalida volleyed into the net.
But Ecuador were awarded a penalty kick when goalkeeper Gabriel Arias clattered Jhegson Mendez, allowing Enner Valencia to level from the spot in the 26th minute.
Sanchez, who had ended a barren five-month spell by scoring in Monday’s 4-0 win against Japan, put Chile ahead early in the second half.
He did so by expertly guiding a bouncing cross into the net with one touch.
Ecuador rarely threatened again and saw defender Gabriel Achilier receive a straight red card in stoppage time for a rash tackle on Arturo Vidal, who had to be taken off.
Chile, Copa America champions in 2015 and 2016, lead Group C on six points.
Uruguay are second on four with Japan third on one and Ecuador bottom with no points.