Elsewhere in Group B, Colombia clinched the top spot with six points with a 1-0 win over Qatar.
Argentina had been second best in the first half of their 2-0 defeat by Colombia last Saturday, but then still conceded both goals in a much improved second-half showing.
Paraguay, who also threw away the lead to draw 2-2 with Qatar in their first game, are second on two points and need to beat Colombia to be guaranteed a spot in the quarter-finals.
"We will try to compete against a great team with the best player in the world".
The 41-year-old experimented with more than 50 players in nine games before the Copa, including trying out eight goalkeepers.
The Paris Saint-Germain forward had been hoping to celebrate reaching 100 worldwide caps during the Copa but the day after that damaging loss, he was sent to train with the substitutes while his replacement Rodrigo de Paul stayed with the first team.
Failure to get out of the group at the Copa America would be a new low, particularly given third place could be enough to see them progress.
The forward admitted he could have continued if Scaloni had not replaced him with Angel Di Maria.
Aside from the football on show, the Copa America 2019 can often spring up other curious moments. After a video review saw a handball by Ivan Piris inside the area, Messi calmly converted from the spot to even the match at the Mineirao. Though it was saved by Franco Armani in the Argentina goal, it was a reminder that Otamendi's status as a back-to-back Premier League champion may be Pep Guardiola's most underrated achievement to date.
Argentina still sits at the bottom of Group B, despite the 1-1 draw Wednesday in Belo Horizonte, Brazil.
"We had a very clear plan but couldn't execute it as we kept losing the ball". A poor crowd of just over 22,000 people watched the game on a chilly night.
Argentina coach Lionel Scaloni went for broke at half-time, bolstering his attack with the introduction of Sergio Aguero and the move paid instant dividends.