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The African side, playing in its first FIFA World Cup, had twice come from a goal down to silence the vast majority of the crowd of 64,478 packed into a hot and humid Miami Stadium.

Six minutes into the second period of extra-time, Lionel Messi swung a corner into the box and Cristian Romero rose to head home off the arm of Cape Verde centre back Diney Borges and finally set up a date with Egypt in Atlanta next Wednesday (AEST).

Messi had, almost inevitably, given Argentina the lead in the 29th minute with his seventh goal of the tournament but Deroy Duarte equalised just before the hour mark.

The Blue Sharks held on to send the match into an additional half hour before Lisandro Martinez lashed a sumptuous shot into the roof of the net in the second minute of the first period of extra-time to put Argentina ahead again.

Cape Verde was not done yet, however, and left back Sidny Lopes Cabral curled a stunning shot into the top corner of the net in the 103rd minute of the contest to put the scores back on level terms at 2-2, sparking wild celebrations from his team on and off the pitch.

Lopes Cabral could have equalised again after Romero's goal but his finely struck free-kick was saved by Argentina goalkeeper Emiliano Martinez, who had to be at his best to deny Cape Verde in the dying minutes.

Cape Verde was beaten but far from outclassed by the three-time world champion as it put in a fourth magnificent display of teamwork and grit at its first FIFA World Cup.

The only one of the four FIFA World Cup debutants to make it to the Round of 32 and ranked No.67 in the world coming into the tournament, Cape Verde had hoped to frustrate Argentina as it did Spain, Uruguay and Saudi Arabia in group-stage draws.

It succeeded, while showing no shortage of quality of its own, for much of the game with a never-say-die desperation and a neat pass-and-move game.

Messi aside, Argentina was largely bereft of ideas against an obdurate defence and Cape Verde libero Kevin Pina was the most impressive player on the park for long periods of the contest.

It was Argentina that made the breakthrough in the 29th minute, however, when Lisandro Martinez lofted a long ball over the top of the defence to the feet of Messi.

The 39-year-old maestro took a touch with the outside of his left boot and buried it in the roof of Vozinha's net for his 20th goal over six editions of football's global showpiece.

Cape Verde knew it would need to score to keep its campaign alive and Duarte fired a shot at goal soon after half-time that drew a diving save out of Martinez.

Just before the hour mark, captain Ryan Mendes was freed down the right and his pass into the box found the Dutch-born midfielder, who controlled the ball with his left foot before drilling it past Martinez with his right.

Messi had a chance to put Argentina back in front four minutes later when he was played through on goal, but Vozinha stood up well to keep his shot out of the net.

One of Messi's trademark free-kicks was tipped away by Vozinha in the 72nd minute and Cape Verde defender Pico Lopes had to intervene to prevent Enzo Fernandez from scoring 10 minutes later.

Cape Verde held on to force the dramatic period of extra-time and will now return home as heroes having put the tiny island-nation firmly on the footballing map.