Lionel Messi broke the deadlock in the 29th minute, giving the reigning champions the lead they had threatened from the opening exchanges.
Cape Verde, appearing in their first World Cup, refused to be overawed by the occasion.
Deroy Duarte levelled for the underdogs in the second half, dragging the match into extra time and setting up a grandstand finish.
Argentina appeared to have snatched the lead again when Lisandro Martinez blasted a shot past Cape Verde goalkeeper Vozinha in extra time.
But Cape Verde responded once more, with Sidny Lopes Cabral restoring parity in the 103rd minute.
The decisive moment came late on, when the ball ended up in the net following a passage of play initially credited to Cristian Romero.
That goal was subsequently reclassified as an own goal by Cape Verde, sending Argentina through to the round of 16.
Argentina coach Lionel Scaloni gave his reaction to the dramatic victory afterwards.
"It was a really tough match. You always have to focus on the positives. This team never gives up. And I have to give credit to our opponents. Today Cape Verde proved they're a great team. The truth is, everyone finished really tired, because they gave it their all.
"This is Argentina... If you don't understand that, it's hard. We Argentines will understand it, because nothing comes easy."
Cape Verde's run in North America ends at the Round of 32 stage, but their performance against the reigning world champions will do little to diminish the achievement of reaching the knockout rounds in their debut tournament.
Argentina, by contrast, move on with questions still to answer.
Scaloni's side struggled to see off a side ranked far below them in the world rankings, needing two moments of fortune in extra time to avoid a shoot-out or elimination.
The manner of the winning goal, initially awarded to Romero before being amended to an own goal, will do little to settle nerves in the Argentina camp ahead of tougher opposition to come.
Cape Verde's players and staff received warm applause from neutral supporters inside the stadium as they left the pitch, their tournament ending with heads held high.
For Argentina, the focus now turns to preparing for their next assignment as they continue the defence of their world title.
Scaloni's message to his players and supporters was clear: nothing in an Argentina shirt comes without a fight.