Published: 2019-06-20 10:09:09 BdST
Richard Sanchez put Paraguay ahead in the 37th minute, firing into the bottom corner after a lung-bursting run from Miguel Almiron down the left wing.
Argentina captain Messi levelled in the 57th minute after a handball by Paraguay's Ivan Peris which no-one on the pitch appealed for but was spotted by a VAR review.
Paraguay were awarded a spot-kick of their own a few minutes later but Derlis Gonzalez's shot was saved by Argentina keeper Franco Armani, who was lucky to avoid a red card when he fouled Gonzalez outside his area at the end of the first half.
Argentina are still bottom of Group B with one point from two games but they could progress to the last eight with a win over third-placed Qatar in their final group game on Sunday.
The result ensured Colombia clinched top spot in the group on six points after they beat Qatar 1-0 earlier on. Paraguay are second on two points.
Argentina's caretaker coach Lionel Scaloni made four changes to the team that were beaten 2-0 by Colombia, which included dropping Sergio Aguero. While they were more solid, they lacked creativity throughout.
Their best move came when substitute Aguero chested down a cross and laid it off to Lautaro Martinez who hit the crossbar on the volley. The ball fell to Messi but the captain's shot was tipped away by Paraguay keeper Roberto Fernandez.
Messi was preparing to take the resulting corner when the referee went to consult the screen and then ruled that Martinez's shot had struck the outstretched hand of Peris.
The Barcelona forward, who is searching for his first trophy with Argentina after losing four major finals, blasted in from the spot but his goal threatened to be cancelled out when Nicolas Otamendi fouled Gonzalez with an awfully timed tackle.
Yet Argentina enjoyed a third slice of good fortune as Armani came to the rescue by punching Gonzalez's spot-kick away, and the River Plate goalkeeper also saved a later free kick from Matias Rojas.
"I think we improved the poor image we left after the first game," Argentina striker Martinez told reporters.
"We have a very difficult game ahead of us to qualify but we'll give it our all to reach the next round."