Angel Romero's penalty gave Paraguay a first-half lead, but his opener was cancelled out by Nicolas Gonzalez's header.
After falling behind, Argentina were dominant, and Lionel Messi had a second-half goal ruled out following a VAR check.
Lionel Scaloni's men were forced to settle for a point and are unbeaten through three qualifiers to top the CONMEBOL standings with seven points, while Paraguay are also yet to taste defeat and have five points.
After a bright start, Paraguay took a deserved 21st-minute lead through Romero's penalty.
Miguel Almiron's strong run at the Argentina defence saw the attacker brought down by Lucas Martinez, and Romero stepped up to place the spot-kick into the bottom corner.
Argentina started to enjoy more possession but struggled to create much of note before Gonzalez equalised prior to half-time thanks to his first international goal.
Gonzalez met a good corner from Giovani Lo Celso, who replaced the injured Exequiel Palacios during the first half, to head in at the near post and make it 1-1.
The equaliser gave Argentina a boost and Rodrigo De Paul's long-range drive forced a save from Paraguay goalkeeper Antony Silva on the stroke of half-time.
Lautaro Martinez and Paraguay defender Gustavo Gomez squandered good chances shortly after half-time before Argentina thought they had the lead.
After a cutback from Lo Celso, Messi beat Silva despite a strong hand from the Paraguay goalkeeper, but the goal was ruled out for a foul in the build-up after a VAR check.
Messi went close in the 72nd minute, Silva tipping his free-kick onto the crossbar after Lo Celso was fouled as Argentina continued to push, but they were unable to find a winner.
What does it mean? Argentina extend unbeaten run
Now unbeaten in three World Cup 2022 qualifiers, Argentina also stretched their overall run to 10. They made a slow start in Buenos Aires but were dominant after falling behind and deserved all three points. Argentina were denied a third straight win to begin a qualifying campaign, something they have managed just three times before.
Lo Celso provides spark
Palacios' injury was unfortunate for Argentina, but Lo Celso's introduction just before the half-hour mark boosted Scaloni's side. The Tottenham attacker assisted Gonzalez's equaliser and set up Messi's disallowed goal, while he was a constant threat.
Paraguay continue to trouble Argentina
Since being belted 6-1 by Argentina at the Copa America in 2015, Paraguay have gone four games unbeaten in meetings between the countries. In World Cup qualifying, Paraguay have won three and lost just two of their past eight clashes with Argentina.
Argentina make the trip to Lima to face Peru on Tuesday, while Paraguay host Bolivia on the same day.