With the scores drawn at 1-1 after extra-time, Neymar’s penalty in the shootout proved decisive when Chile defender Gonzalo Jara smashed the final spot-kick against the post.
Julio Cesar saved two penalties in the shootout as the hosts booked a last-eight encounter with Colombia, who were 2-0 winners over Uruguay.
Brazil were pushed to the very end by Chile, who could have won the absorbing contest in Belo Horizonte had Mauricio Pinilla’s shot in the last minute of extra time gone in rather than hit the crossbar.
Scolari paid tribute to Jorge Sampaoli’s side and insisted his own team showed tremendous spirit to remain in the competition.
He said: “We knew when the draw was made we were going to be in for a tough time. They’re a very organised team with a good tactical system and fine players.
“Every credit to Chile but this penalty shootout win says a lot about us and about our desire to progress.”
Brazil had taken the lead through a goal awarded to David Luiz - even though it appeared to come off a Chile player - before Chile netted a deserved equaliser through Alexis Sanchez.
Brazil’s win also saved English referee Howard Webb from becoming the country’s public enemy number one after he found himself at the centre of huge controversy when he disallowed a goal by Brazil forward Hulk for handball.
Sampaoli believes his side were unlucky to bow out of the tournament and can be proud of their efforts.
He said: “I’m not happy. We gave everything we had and we were up against a whole stadium, against the hosts and we didn’t go through.
“Our players made history and they did their country proud. We had possession of the ball, we put in a lot of effort, we took more risks and we fought till the end.”
Colombia are through to the World Cup quarter-finals for the first time in their history thanks in no small part to James Rodriguez, whose inspired performance was lauded by coach Jose Pekerman and Uruguay counterpart Oscar Tabarez.
In the build-up to the tournament Los Cafeteros were hit by an injury to star striker Radamel Falcao, yet the 22-year-old has filled the void left by his Monaco team-mate with aplomb.
Rodriguez continued his exquisite form by netting twice in Saturday’s last-16 tie against Uruguay at the Maracana, where he scored arguably the individual goal of the tournament before netting one of the best team efforts in a 2-0 win.
Those goals lifted Rodriguez to the top of the World Cup scoring charts and set up a mouth-watering clash between Colombia and hosts Brazil in Fortaleza next Friday.
“In my long experience in football, I’ve had in my teams extraordinary footballers, elite players of a very high technical level,” Pekerman said when asked about Rodriguez, named man-of-the-match for the third time in four matches.
“I place everything in James because I saw the extraordinary conditions that he had.
“What is most surprising is that at his young age he does not have any objection, any problem in being responsible for doing things that football players take many years to understand.
“Football forces you to deal with a lot of situations, you have to handle these situations and we have a technical player who has every attribute of a top-notch player at a world level.
“I greatly praise what he has been doing and I never had any doubts that this was going to be his World Cup. James Rodriguez’s World Cup.”
Those comments were echoed by Uruguay coach Tabarez, who believes Rodriguez has been the best player in Brazil.
“Diego Maradona, Lionel Messi, Luis Suarez and James Rodriguez - they do things because they have a certain gift which makes them special,” he said.