But it was Paraguay's Richard Sanchez who opened the scoring to put his side ahead in the 37th minute.
Minutes after Messi's equaliser, Paraguay were awarded a spot-kick of their own but Derlis Gonzalez's effort was punched away by Argentina keeper Franco Armani, who narrowly avoided a red card when he fouled Gonzalez outside his area at the end of the first half.
Now, with one point to their name, Argentina are on the verge of a crisis, and they must win against Qatar to have any hopes of progressing to the knockout rounds.
Earlier, Colombia topped Qatar, 1-0, in a pretty evenly-played game.
The 2019 Copa America isn't going much better for Argentina than the 2018 World Cup, with a slow defense and a lack of playmakers in the middle of the park keeping the Albiceleste from making a real difference in the tournament.
The 25-year-old speed machine can play as a winger or in the number 10 role and he's already a big favourite at St James' Park.
Angel Di Maria looks set to be dropped against Paraguay after a bad display in the previous game. The two best third-placed teams advance to the quarter-finals, meaning Argentina will be knocked out of the tournament with anything less than a win against Asian Cup winners Qatar.
Midfielder Roberto Pereyra and defender Milton Casco also joined the starting lineup, replacing Guido Rodriguez and Renzo Saravia. A handball from Paraguay's Ivan Piris led to Messi's second half equalizer.
Otamendi floored Gonzalez in the box on 62 minutes but the same player then saw the resulting penalty saved by Argentina's Armani.
"It would be insane if we can't advance from the group stage, when basically three teams advance from each group". They are, however, yet to win a game in this edition and will be looking to record their first victory so as to advance to the next phase.
"I could have continued playing but the coach made a decision to take me off".
"I think we improved the poor image we left after the first game", Argentina striker Martinez told reporters.