In what was a largely scrappy and tight encounter, two-time defending champion Chile was held to a 0-0 draw at Arena Corinthians in Sao Paulo, with VAR (video assistant referee) playing its part in the scoreline.
Chile twice thought it had found a breakthrough, but strikes from Charles Aranguiz and Arturo Vidal were both ruled out after reviews.
But La Roja delivered in the penalty shootout, Sanchez converting the winner to seal a 5-4 success after William Tesillo missed Colombia's fifth kick.
Chile edged the first half and Aranguiz forced a superb save from Colombia goalkeeper David Ospina with a 12th-minute header, although the VAR would likely have ruled out the effort for offside if it had gone in.
Aranguiz thought he had given his side the lead just four minutes later.
Sanchez played in Jean Beausejour down the left and Ospina, along with Davinson Sanchez, made a mess of the cross, with Aranguiz netting from an angle.
However, the goal was ruled out after a VAR review, with replays showing Sanchez was narrowly offside down the left.
The first half lacked any real rhythm, with Guillermo Maripan heading a Beausejour cross over 10 minutes before the break and Vidal curling a Sanchez pass just wide moments later.
The battle continued after the break, with efforts from distance or set-pieces appearing the most likely avenues to a goal for both teams.
VAR was again involved after Chile thought it had its opener through Vidal in the 71st minute.
Vidal finished clinically from inside the area, but the goal was ruled out after Sanchez's volley hit the arm of Maripan in the build-up.
That was the last of the opportunities through 90 minutes before Chile was too good in the shootout.
Vidal, Eduardo Vargas, Erick Pulgar and Aranguiz produced fine spot-kicks before Tesillo fired wide, Sanchez stepping up to put away the winner.
Chile will face either Uruguay or Peru in a semi-final in Porto Alegre on Thursday (AEST), while Colombia looks ahead to friendlies with Brazil and Venezuela in September.