Five-time Ballon d’Or winner Messi’s 57th-minute penalty cancelled out a first-half opener from Paraguay’s Richard Sanchez on Wednesday night.
But Argentina also had goalkeeper Franco Armani — perhaps lucky to still be on the field at that point after denying Derlis Gonzalez a clear goalscoring opportunity in the first half — to thank for saving a second-half penalty.
“We have to think positively and keep our confidence,” said Barcelona forward Messi.
“We need to improve and arrive in the best shape possible for the next match … and we have to win.”
Defeat would not have ended Argentina’s hopes of reaching the quarter-finals since eight of the 12 participating teams will progress to the knock-out stages.
As well as the top two in each of the three groups, the best two third-placed finishers qualify.
“It would crazy if we couldn’t progress from the group given that there are practically three teams that go through,” said Messi, who will turn 32 on Monday.
Defeat at the Minerao stadium in Belo Horizonte would have meant that Argentina could finish no higher than third — Colombia have already won the group after two straight wins.
Now they know that victory over invitees Qatar in Porto Alegre on Sunday will likely be enough to see them into the quarters.
However, the Asian champions are in the same boat as Argentina, with one point from two matches, and are even ahead of Lionel Scaloni’s team on goal difference.
And Argentina have showed few signs of finding the kind of form that took them to three major tournament finals in as many years — all lost — from 2014-16.
Searching for a team
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.
Scaloni dropped Manchester City’s Sergio Aguero and Angel Di Maria of Paris Saint-Germain for the Paraguay match but Argentina produced another disjointed and unconvincing display.
But Messi insisted that the level of competition is tougher than some appreciate and pointed to Venezuela holding Brazil to a 0-0 draw on Tuesday as proof.
“We knew it wasn’t going to be easy playing in a Copa America, we’re searching for a team, for a performance,” said Messi.
“Venezuela made it hard for Brazil. At the moment we’re not winning any more in the (Argentina) shirt and we have to put things right to earn the three points” against Qatar.
Messi came to Brazil hoping to end a painful run of losing finals with Argentina.
Despite winning La Liga 10 times and the Champions League four times with Barcelona, he’s never won a major honor with his country.
Three times — 2007, 2015 and 2016 — he’s lost in a Copa America final and in 2014 Argentina were also defeated by Germany in the World Cup final.
But after their first two performances here, the question is not whether they can end their 26-wait for a major title, but simply whether they can avoid taking the first flight home.