The Chile forward, who scored just two goals in the whole of last season for United, netted his second in four days to secure their place in the next round.
United's £505,000-a-week man played through pain in the Group C clash against Ecuador after suffering a first-half ankle injury.
Former West Ham forward Enner Valencia levelled the score from the penalty spot after Jose Pedro Fuenzalida's eighth minute strike.
But six minutes into the second-half Sànchez popped up at the far post and saw his scuffed effort bounce off the turf and into the net.
The 30-year-old celebrated with wild relief after a miserable season at Old Trafford which was blighted by poor form and a knee injury in March.
"When I play for my country, I'm always playing happy," Sànchez told reporters.