The two teams could not be separated over the 90 minutes, with VAR twice coming to Colombia's rescue after the defending champions thought that they had made the breakthrough.
Alexis Sanchez scored the decisive spot kick, though, to ensure that La Roja would keep alive their hopes of winning a third straight Copa America title.
Kickoff was delayed for 20 minutes as Chile arrived at the stadium late due to local traffic. The defending champions also started slower on the field as Colombia enjoyed some positive early moments in Sao Paulo.
Gary Medel and Arturo Vidal had both been declared fit ahead of kickoff and lined up for La Roja, while Radamel Falcao was given the nod at the tip of the Colombian attack despite Duvan Zapata's two goals this summer.
Roger Martinez had a goal-bound strike blocked by Medel in the sixth minute as Colombia continued to threaten, but Chile were also offering a threat down the other end, with Alexis Sanchez - so disappointing for Manchester United last season - looking energised in the final third.
Charles Aranguiz was key to Chile's run to the quarter-finals and the number 20, who picked up an early booking, brought a smart save from David Ospina in the 13th minute after ghosting into the Colombian box.
Aranguiz was not to be denied three minutes later, though, with the midfielder converting from close range after a terrible mix-up involving Davinson Sanchez and Ospina. A lengthy VAR check followed, though, and the goal was eventually disallowed for an offside involving Alexis Sanchez.
Falcao tested the gloves of Gabriel Arias in the 25th minute as Colombia looked to register, but chances were few and far between in the period that followed the disallowed goal.
Both sets of players were involved in a pushing match in the 32nd minute with Yerry Mina and Alexis Sanchez both involved, but the referee was quickly on the scene to calm things down.
Chile had a super chance to make the breakthrough in the 36th minute when Guillermo Maripan climbed highest inside the Colombia box to meet a cross from Jean Beausejour, but the centre-back missed the target.
There was certainly a bite to a number of challenges that were occurring - just two bookings arrived in the first period, however, and there are no goals following the decision to reverse Aranguiz's 16th-minute effort.
Vidal had the final opportunity of the first half after smart work from Alexis Sanchez, but the Barcelona midfielder sent his effort over Ospina's crossbar from the edge of the box.
James Rodriguez sent a free kick into the side-netting of the Chilean goal in the first big moment of the second period, which needed a goal if a penalty shootout was to be avoided.
Eduardo Vargas then forced Ospina into a smart save down the other end as both teams had opportunities to make the breakthrough in the first seven minutes of the second half.
The score was still level approaching the hour - half-chances were arriving, though, and Cuadrado just failed to gather a James Rodriguez pass inside the Chilean box after an excellent piece of skill from the Juventus midfielder.
Alexis Sanchez was next to try his luck as the Chile attacker broke into a dangerous position before sending a low strike just wide of Ospina's post in the 61st minute.
Chile had the ball in the back of the net for a second time in the 71st minute through Vidal, but the midfielder's effort was eventually chalked off for a handball as VAR again came to Colombia's rescue.
Duvan Zapata eventually replaced the ineffective Falcao in the 77th minute as Carlos Queiroz's side looked to force the issue in the latter stages of normal time.
La Roja were the team on the front foot, though, and Erick Pulgar missed Ospina's crossbar with a header in the 84th minute during a strong spell of pressure.
There was to be no late drama in Sao Paulo, however, with a goalless 90 minutes meaning that the South American rivals needed a penalty shootout to decide which team would advance to the semi-finals.
James Rodriguez got Colombia off to the perfect start by picking out the bottom corner with a low effort, but Vidal answered straight back with an emphatic finish.
The first two spot kicks were a sign of things to come as the next six players all registered, but William Tesillo missed Colombia's fifth and final effort, which allowed Alexis Sanchez to convert and send the defending champions into the final four.
Chile, who have joined Brazil and Argentina in the latter stages of the competition, will face either Uruguay or Peru in the second semi-final next week.
COLOMBIA (4-3-3): Opsina; Tesillo, Sanchez, Mina, Medina; Cuadrado, Barrios, Uribe (Cardona 67'); James, Falcao (D Zapata 77'), Martinez (Diaz 81')
CHILE (4-3-3): Arias; Isla, Medel, Maripan, Beausejour; Aranguiz, Pulgar, Vidal; Fuenzalida (Pavez 75'), Vargas, Sanchez