Messi converted a 57th-minute spot-kick after Argentina had fallen behind to a Richard Sanchez goal in Belo Horizonte.
Argentina goalkeeper Franco Armani was needed to keep his side level shortly after the hour-mark, though, saving a Derlis Gonzalez penalty.
The draw saw Argentina earn ITS first point in Group B, but they remain bottom behind Colombia, Paraguay and Qatar.
While Argentina enjoyed the better start, the game lacked any rhythm in the opening 20 minutes after numerous fouls and stoppages.
It was Paraguay that created the best chance of the opening half-hour, Gonzalez's strike deflected just wide after a counter-attack.
Messi had an opportunity for Argentina in the 34th minute, but the Barcelona star's free-kick from 30 yards lacked the power to trouble Paraguay goalkeeper Roberto Fernandez.
Paraguay would break the deadlock before half-time through Sanchez's first international goal.
Newcastle United midfielder Miguel Almiron set up the goal with a brilliant run down the left, before crossing for Sanchez to finish into the bottom corner from inside the area.
Lautaro Martinez hit the crossbar for Argentina six minutes into the second half after good work from half-time substitute Sergio Aguero, but it was that moment that led to the equaliser.
Argentina was awarded a penalty after a VAR review, with replays showing Martinez's strike hit the arm of Ivan Piris.
Messi stepped up, putting the penalty into the bottom corner for his 68th international goal.
However, Paraguay was awarded a spot-kick of its own after a poor challenge by Nicolas Otamendi on Gonzalez, but Armani dived low to his left to push away the penalty in the 63rd minute.
Martinez headed over soon after for Argentina and both teams struggled to create chances late on as they shared the spoils.