It took longer than Argentina may have hoped, but advancement came for them nonetheless in a tight affair against Ecuador.
The tense match featured Argentina opening the scoring when Lisandro Martinez nodded home a headed pass from Alexis Mac Allister in the first half.
But on the doorstep of full time, Kevin Rodriguez equalized for La Tri to force a penalty shootout.
It was in that shootout that Nicolas Otamendi slotted home a game-winning penalty to see La Albiceleste advance to the semifinals.
There, they will play the winner of Friday's match between Venezuela and Canada, a team Argentina already beat 2-0. Argentine captain Lionel Messi will hope to add more to his decorated trophy cabinet and this win will see him get one step closer to that task.
Lionel Messi and Argentina are advancing to the semifinals of the 2024 Copa America
A headed goal from Lisandro Martinez (25) was all that was required of La Albiceleste
Through the first 20 or so minutes, Ecuador looked like they had momentum on their side over the reigning champions of the tournament.
Argentina's breakthrough came off a corner when Enzo Fernandez's shot was blocked and went over the end line.
The corner whipped in by Messi was touched on by Alexis Mac Allister - perfectly played straight behind the Liverpool midfielder into open space.
That area was immediately filled by an oncoming Lisandro Martinez, who cleanly headed the ball into the net for a 1-0 Argentina lead.
But even at halftime, Ecuador held a slight advantage over Argentina in expected goals (0.50-0.49) despite holding less possession.
Bodies flew just five minutes into the second half as Messi, Moises Caicedo, Christian Romero, and one other Argentina player hit the deck.
After a lengthy pause, Argentina tried doubling their advantage - but Lautaro Martinez saw his shot go wide.
A sliver of light came for Ecuador in the 60th minute when Alan Franco was shoved down in the box by Rodrigo De Paul and referee Andres Matonte whistled for a penalty.
Ecuador's 34-year-old captain Enner Valencia stared down Argentina keeper Emi Martinez and could not survive the Argentine mind games. His slow-rolling shot sent the keeper the wrong way but clattered off the post.
Rodrigo De Paul (7) raises his hands pleading innocence after giving away a penalty
Enner Valencia (13) grimaces after he missed a penalty against Argentina at the Copa America
Emiliano Martinez (L) and Lisandro Martinez (R) celebrate after a missed penalty from Ecuador
On the doorstep of full time, Kevin Rodriguez (11) kept Ecuador alive - forcing penalties
Martinez turned to the crowd and pumped his arms in victory as Valencia was left to shake his head. The largely pro-Argentina crowd chanted his nickname 'Dibu' in salute.
It was at that point where the momentum of the game turned nearly fully to La Albiceleste - who out-possessed their opponents through the first 70 minutes.
A Messi free kick led to a counterattack from Ecuador - but despite a pair of crosses, including one that fell to Valencia, they didn't manage a single shot on target.
It seemed destined that Argentina would hold on to maintain their victory in front of throngs of fans in Houston as five minutes were added on.
But Ecuador fought back. After a clattering of players, La Tri took a quick free kick. In desperation, a cross was sent into Kevin Rodriguez - who put a head on it.
The ball sailed across the diving body of Emi Martinez and into the net to equalize in the dying seconds.
The match immediately went to penalties - raising the heartrates of everyone at NRG Stadium in Houston.
Lionel Messi tried a cheeky Panenka for his penalty, but his shot hit the bar and went over
He was redeemed by Emi Martinez - who saved two shots in the shootout to save the day
It was Nicolas Otamendi who stepped up to take the game-winning penalty kick for Argentina
Up first was Messi, who seemed calm under pressure - but did too much. His Panenka attempt hit off the crossbar and went over for a miss.
Emi Martinez redeemed his captain with an opening round save on Angel Mena to send Argentina fans into rapture.
Substitute Julian Alvarez put his attempt over the outstretched arms of keeper Alexander Dominguez before Martinez stepped up for Argentina once again - putting a glove on the shot from Ecuador's Alan Minda.
Round three started with Alexis Mac Allister calmly slotting his shot into the right side netting as John Yeboah finally scored the first Ecuador penalty of the night.
Gonzalo Montiel, the man who won Argentina the World Cup, calmly slotted his away - forcing Ecuador into a do-or-die shot at the boot of Jordy Caicedo. Caicedo booted his attempt down the middle to keep his nation alive.
With the chance to advance, up stepped Nicolas Otamendi. He snuck his shot past the diving Dominguez and Argentina advanced to the semifinals at MetLife Stadium in New Jersey.