Copa America winners Uruguay lead the nine-nation South American group with seven points from three matches, followed by Colombia, who have a game in hand, Argentina, Paraguay and Venezuela.
Colombia let three points slip away at home when Venezuela scored a late equaliser for a 1-1 draw, while Paraguay snatched their first win, 2-1 at home to Ecuador.
Uruguay's Suarez said it had been hard to contain his emotions after the four-goal performance.
"You want to burst into tears, your stomach turns," he said. The striker came off in the 77th minute to a standing ovation at the Centenario in Montevideo.
Suarez has scored 26 for Uruguay, six short of the record held by fellow forward Diego Forlan, who missed the match through injury. Brazilian Romario was the last South American to score four goals in a qualifier, against Venezuela in 2000.
"I have a great admiration for how he went overcoming challenges in his career," said Uruguay coach Oscar Tabarez.
Chile coach Claudio Borghi, who kicked five players out of his squad for indiscipline in midweek, said the incident was in the past.
"The week was not simple but what happened up to this match is history. We're going to talk about how they felt," he told the post-match news conference.
Chile next host Paraguay on Tuesday, while Uruguay will play Italy in a friendly in Rome.
Messi, who has no problem scoring for club side Barcelona, must find a way to link up with his Argentina team mates if they are to reach the 2014 World Cup finals in Brazil.
"We can't let another single point escape us, we're angry because we had chances to score but we were unlucky," a visibly irritated Messi told reporters.
Argentina had much more of the ball and by far the greater number of chances but were frustrated by a packed, well-organised Bolivia defence.
They also had to fight back from a goal down after Marcelo Martins latched onto an error by defender Martin Demichelis and scored in the 55th minute.
Ezequiel Lavezzi grabbed the equaliser in the 59th minute shortly after coming on as a substitute.
It was Bolivia's first World Cup qualifying point in Argentina in nine games.
"The important thing was to take precautions with Messi because he's a player who, if you don't control him, he'll win the match... so we gave him little room," Bolivia coach Gustavo Quinteros said.
Argentina next play Colombia, who let two points slip from their grasp in a 1-1 draw in Barranquilla with northern neighbours Venezuela.
The home side went ahead with an early goal from winger Fredy Guarin but squandered a string of chances and paid the price when Venezuela substitute Frank Felstcher equalised 12 minutes from time.
Paraguay had to work hard for their first win, 2-1 at home to Ecuador, breaking the deadlock thro
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