The Argentinians will play Real Madrid in Sunday's final while Auckland City face Cruz Azul (Mexico) in the playoff for third place.
But it was another incredible effort from Auckland, one that will be remembered for years to come.
Auckland City weren't given much hope before this match. They had overachieved and this would surely be a bridge too far and end in a convincing victory to the South American side. It was going to script when San Lorenzo took the lead right on the stroke of half time, a deflating time to concede at any level. But this team won't lie down and Berlanga's goal brought them back into the match.
It was a cagey start from both sides. Auckland City were always going to take a cautious approach - knowing that an early goal to the Argentinians would be fatal - but it was a surprise to see San Lorenzo also reluctant to press forward and attack in numbers.
This tournament - and its many predecessors, dating back to the 1970s - has always been cherished by the South Americans, who see it as a true world championship. And San Lorenzo had been the nearly team of Argentina, until last year the only one of the big five (Boca Juniors, River Plate, Independiete and Racing are the others) to not win the Copa Libertadores (South American Championship).
That win was their ticket to Morocco and they weren't going to take any chances, especially with the carrot of a once in a lifetime match against Real Madrid.
San Lorenzo created some opportunities in the first half hour - Tamati Williams was forced into a good near post save - but Auckland City held their own and didn't look out of place on their biggest stage yet.
Indeed, Tade twisted away from two defenders to create a half chance but hit his shot straight at the goal keeper, while Ryan De Vries failed to get another half chance on target.
At times Auckland City could have shown more attacking intent but they mostly got the balance right. They kept the ball when they could, pressed whenever possible and closed down the spaces quickly for the Argentinian.
The plan was working but unfortunately, it all came unstuck moments before halftime - the worst possible time to concede - as the Auckland side switched off for a moment and Pablo Barrientos struck a fine shot just inside the near post. By the standards of their defending at this tournament, it was a soft goal to concede.
Auckland, as they had to, took a more direct approach in the second half. San Lorenzo still looked the more dangerous team, but the Oceania side began to find more space on attack.
Still the Argentinians looked comfortable - until Berlanga's brilliant goal brought their back into the contest. A lovely pass by Tade prised the defence open and the Spaniard scored from a near impossible angle, after the ball had eluded both the goal keeper and Ryan De Vries.
San Lorenzo were stunned and were unlucky not to retake the lead as Martin Cauteruccio hit the post with a spectacular volley from the edge of the area.
But astonishing as it seems, Auckland had an even better chance following a quick break in the 77th minute.
De Vries drew the last defender and squared for Tade, who only had the goal keeper to beat. But just like he did in the first match, Tade's first touch let him down and his shot squeezed tamely wide.
Any hopes they could hang on in extra time were dashed in the 94th minute, as replacement striker Mauro Matos struck after the ball broke to him just inside the area.
San Lorenzo 1 (P. Barrientos 45+2, M. Matos 93)
Auckland City 1 (A. Berlanga 67
Halftime: 1-0
