Paraguay stunned the twice world champions with a fast break from deep in their own half, Angel Romero playing the ball into the path of Derlis Gonzalez who beat Sergio Romero with a shot inside the near post in the 18th minute.
The defeat left Argentina in fifth place in the South American group with 16 points, one point ahead of Paraguay and four behind leaders Uruguay, who were overtaken by Brazil after beating Venezuela 2-0 away.
Ecuador are third on 17 points, level with Colombia, who are in fourth on goal difference.
The top four go through to the 2018 finals in Russia while the fifth placed team qualifiers for an intercontinental playoff for another berth.
Argentina will be desperate to have a fit Messi back for their next qualifier away to Brazil in Belo Horizonte on November 10.
They could have got back on level terms two minutes into the second half but Paraguay captain Justo Villar saved Sergio Aguero’s weak penalty.
Uruguay were held to a 2-2 draw away to Colombia with Luis Suarez scoring his 19th goal in qualifying matches, equalling the record held by former Argentina striker Hernan Crespo.
The Uruguayans were pegged back in steamy Barranquilla by Colombia striker Yerry Mina’s late headed equaliser after Suarez had put them ahead in the 73rd minute.
Midfielder Cristian Rodriguez had headed a 27th-minute leveller for Uruguay to cancel out Abel Aguilar’s opening goal.
Ecuador’s Enner Valencia scored twice, including an 89th minute equaliser, when they were a man short as they came from two goals down to draw with Bolivia in La Paz. Midfielder Luis Caicedo was sent off 12 minutes from time for a second booking.
Pablo Escobar scored both goals for Bolivia.
Copa America holders Chile, who had not won in three games, produced a 2-1 victory over arch-rivals Peru in Santiago that keeps them in seventh place on 14 points, one behind Paraguay.
Arturo Vidal scored both Chile’s goals, the winner in the 85th minute after Edison Flores’s equaliser.