Juan Arce and Marcelo Martins scored either side of halftime to give Bolivia a deserved victory over an Argentina side who defended poorly throughout an exciting end-to-end encounter.
The defeat in the high altitude of the Andean city, coupled with Colombia's 2-0 win away in Ecuador and Chile's 3-1 defeat of Venezuela, means the twice World Cup winners fall from third to fifth in the 10-team South American qualifying group.
The top four teams qualify automatically for Russia 2018 and the fifth-placed side goes into a playoff with a team from Oceania.
Argentina have 22 points, one fewer than Chile, two behind Colombia and eight in arrears to group leaders Brazil. Uruguay, who play Peru later on Tuesday, are also above them on 23 points.
Argentina were already without suspended regulars Javier Mascherano, Lucas Biglia and Gonzalo Higuain and their task was made even tougher when captain Messi was handed a four-match ban for swearing at a match official hours before the game started.
Bolivia appeared fired up by the decision and got the goal their early pressing deserved through Arce after 31 minutes.
The skinny winger outmuscled Facundo Roncaglia on the penalty spot to head home a deep cross from Pablo Escobar on the right.
Roncaglia was at fault again for the second goal seven minutes into the second half.
The Celta Vigo defender failed to prevent Jorge Flores from escaping to the byline to cross for an unmarked Marcelo Martins.
The big striker had all the time in the world to steady himself and hammer the ball past helpless Argentina goalkeeper Sergio Romero.
Argentina came into the game more when Sergio Aguero replaced Angel Correa 10 minutes into the second half but they could not get a goal.
"We tried to wrestle control of the game from Bolivia (and) the team improved a bit in the second half," said Argentine coach Edgardo Bauza.
"Bolivia won fair and square," he added. "We need to think ahead. We are still alive and we want to qualify for the World Cup."
"We know that this is going to be a battle and we are going to keep fighting."
(Story corrects to remove reference to La Paz as Bolivia's capital.)
(Writing by Andrew Downie; Editing by Toby Davis)
