However, with a rampant Ecuador crushing Paraguay 4-1 in Quito, South America's top three teams remain in charge of their World Cup qualifying chances as the race for the region's fourth direct ticket to the 2014 finals in Brazil hotted up.
Uruguay, Copa America holders and the region's best-performing team at the 2010 World Cup, lost 2-0 in Chile and sank below the qualifying line as their opponents and Venezuela climbed above them into fourth and fifth.
Argentina managed well in the thin air at nearly 4,000 metres in La Paz to draw 1-1 with Bolivia, putting behind them the 6-1 defeat suffered by Diego Maradona's team four years ago.
But they might have won had Messi made the most of a late chance.
Messi stole the ball from a defender and came face to face with Sergio Galarza but a moment's hesitation cost him dear and allowed the goalkeeper to block his shot.
That preserved Bolivia's record as the only South American team to have denied the little ace a goal.
"That's another point we have added and (this result) strengthens the squad (but) we deserved all three," Argentina coach Alejandro Sabella said.
"Obviously we have good chances (of qualifying), we've gone from moderate optimism to greater optimism," he said after taking four points in four days following Friday's 3-0 win over Venezuela in Buenos Aires.
Colombia went to Puerto Ordaz in Venezuela seeking a fifth win in a row after their 5-0 rout of Bolivia at home on Friday but they came away empty-handed as striker Jose Salomon Rondon scored the only goal for the home side.
Ecuador made it six wins out of six home matches in their Atahualpa stronghold nearly 3,000 metres up in the Andes mountains, with two goals from striker Jefferson Montero.
Musical chairs for the top three places has Argentina at home to Colombia on June 7 and visiting Ecuador in Quito four days later after the Ecuadoreans' trip to Peru. Colombia host the Peruvians on their return from Argentina.
Uruguay have a bye on June 7 and will watch the middle positions in the table closely as Venezuela visit Bolivia and Chile travel to Paraguay.
"We have potential (but) we need to know why that potential isn't surfacing," said a worried Uruguay coach Oscar Tabarez, whose team have taken only two points from their last six matches.
"The outlook is much more complicated ... Our objective now is not fourth place, but fifth which gave us our ticket to the last World Cup."
Fifth place, currently occupied by Venezuela, is valid for a play-off with a qualifier from Asia. In 2009, it was against a team from the CONCACAF region - North and Central America, and the Caribbean - with Uruguay beating Costa Rica.
Chile's Argentine coach Jorge Sampaoli said: "We improved in recovering the ball and didn't let a dangerous team like Uruguay, with lots of great players, have a leading role in the match."
Venezuela, who have improved in leaps and bounds over the last decade with a first Copa America semi-final in Argentina in 2011 the highlight, can dare to dream of playing their first World Cup.
They edged into fifth place on 15 points, equal with Chile if behind them on goals scored, leaving Uruguay sixth with 13.
Peru, who have 11 points, can still vie for a berth in their remaining six matches but Bolivia, with nine points, and 2010 quarter-finalists Paraguay with eight, face an almost impossible task to keep pace.