The result means that Paraguay stay second in the section on two points, while Argentina are fourth on a single point, meaning that Lionel Scaloni's side will have to beat Qatar in their final group match on Sunday to stand a chance of booking a spot in the quarter-finals of the competition.
Argentina's team had four changes from the Colombia defeat, with Milton Casco, Roberto Pereyra, Rodrigo de Paul and Lautaro Martinez all coming into the XI. A switch to a 4-4-2 formation saw Messi join Martinez in the final third, while Paraguay lined up in a 4-2-3-1, with Newcastle United's Miguel Almiron operating off the right.
The first 15 minutes passed without incident as both teams struggled to get going in the final third. The stop-start nature of the contest did not help, with Paraguay duo Derlis Gonzalez and Gustavo Gomez both picking up head injuries following collisions with Nicolas Tagliafico and Messi respectively.
The Argentina fans would have been looking for a response following the defeat to Colombia, but it was more of the same from Scaloni's side, who were finding it incredibly difficult to create openings in the final third.
The clock ticked past the 25-minute mark with still no shots from either side. Argentina were showing the greater intent, but Paraguay had two decent openings in the build-up to the 30th minute. First, Gonzalez saw a dangerous cross deflected just wide of the far post, before Junior Alonso fired wide from the resulting corner kick.
Argentina eventually registered a shot on target in the 34th minute from a Messi free kick, but it was a routine save for Roberto Fernandez as Paraguay kept their clean sheet intact approaching the end of the first period.
It got better for Paraguay in the 37th minute, though, when Almiron drove down the left before delivering an excellent cross into Richard Sanchez, who turned the ball into the bottom corner to hand Los Guaranies a shock lead.
The goal was Sanchez's first for his country on just his fourth appearance and Eduardo Berizzo's side immediately went in search of a second, with Gonzalez and Santiago Arzamendia both having attempts in the period that followed.
Argentina goalkeeper Franco Armani then picked up a yellow card in the 45th minute for a rash challenge on Gonzalez, having had a wild swing at the attacker after leaving his goal in an attempt to win the ball. The Paraguay players wanted a red, though, and they had a case due to the nature of the tackle.
Sergio Aguero replaced Pereyra at the interval as Argentina looked for a route back into the match, but Paraguay started the second half the brighter as they searched for a second goal.
Scaloni's side had a golden chance to level in the 52nd minute when Aguero found Martinez inside the penalty box, but the number 22 could only poke his attempt against the crossbar and Messi could not convert from the rebound.
Argentina were ultimately handed a penalty for an Ivan Piris handball, though, with replays showing that the defender blocked Martinez's attempt onto the crossbar. Messi stepped forward for his team and found the bottom corner with a powerful strike to leave the two teams all square in the 57th minute.
Paraguay were awarded a penalty of their own in the 62nd minute when Nicolas Otamendi brought Gonzalez to the ground with a poor challenge. The Paraguay attacker returned to his feet to take the spot kick, but Armani was on hand to make a smart save down low as the score stayed 1-1.
Martinez then missed a glorious chance for Argentina moments later as the striker headed a brilliant Giovani Lo Celso delivery over the crossbar from close range. The striker was replaced by Angel di Maria in the 67th minute as La Albiceleste looked to take advantage of their pressure.
Opportunities dried up in the latter stages, though, with the score still level entering what proved to be seven minutes of additional time at the end of the 90 minutes.
Tagliafico had a goal-bound strike blocked in stoppage time, but there was to be no late drama in Belo Horizonte as Paraguay comfortably defended their lines to pick up an impressive point against an Argentina team that were once again incredibly disappointing on the night.
ARGENTINA (4-4-2): Armani; Casco, Pezzella, Otamendi, Tagliafico; De Paul (Suarez 87'), Lo Celso, Paredes, Pereyra (Agüero 46'); L Martinez (Di Maria 67'), Messi
PARAGUAY (4-2-3-1): Fernandez; Piris, Gomez, Alonso, Arzamendia; Sanchez, R Rojas; Gonzalez (Escobar 90'), M Rojas, Almiron (Ortiz 87'); Santander (Romero 73')