The third draw in a row between the two teams kept Argentina top of the South American group with 24 points from 11 matches. Bolivia have nine points from the same number of games.
The former world champions have now gone four games without a victory against Bolivia since being crushed 6-1 on their previous visit to the Hernando Siles stadium, nearly 4,000 metres above sea level.
With five minutes remaining Messi had an opportunity to win the match for Argentina, stealing the ball from a defender and bearing down on goal with only Sergio Galarza to beat but the goalkeeper blocked his attempt to thread the ball between his legs.
"I hesitated a bit. It's terrible playing at altitude," said Messi who appeared at times to be suffering the effects of the thin air.
"After making a move at speed it's much harder to recover," he told reporters.
Coach Alejandro Sabella, who made eight changes from the team that beat Venezuela 3-0 in Buenos Aires on Friday, said he was pleased with his second-string side's performance.
"It was an excellent display by Argentina. We had some difficulties in the first half but in the last 20, 25 minutes we deserved more," he added.
Bolivia had much the better of the opening exchanges and went ahead in the 25th minute when striker Marcelo Martins rose above defender Hugo Campagnaro to head Alejandro Chumacero's right-wing cross past Sergio Romero.
Argentina, who had a good chance for striker Rodrigo Palacio saved by Galarza with his feet, equalised one minute before half-time.
Messi beat two defenders and fed left-back Clemente Rodriguez whose centre was headed home by midfielder Ever Banega past the diving Galarza.
Romero was the busier keeper in the first half, saving well from Diego Bejarano and an earlier Martins effort from 25 metres while Bolivia's Carlos Saucedo volleyed over from close range.
Ultimately, though, Bolivia had Galarza to thank for their point after two fine second-half saves, first a diving stop from Banega following the midfielder's sharp one-two with his skipper on the edge of the box and then Messi's chance.
Argentina winger Angel di Maria appeared not to have much trouble with the altitude, making a number of trademark weaving runs down the left and going close with a low shot in the 36th minute.
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