Pablo Barrientos put the South American champions ahead with a brilliant volley in first-half injury time – it was the first goal the Kiwis had conceded at Morocco 2014 in their third outing – before Angel Berlanga equalised from a tight angle following excellent work from Emiliano Tade.
Aware Auckland had reached the quarter-finals by winning a penalty shootout, San Lorenzo emerged for the additional half-hour with added urgency, and it quickly paid off when substitute Matos dispatched a loose ball in the box.
Edgardo Bauza's side will return to the Stade de Marrakech on Saturday to face Real, who thrashed Cruz Azul 4-0 in the other semi-final.
Earlier that day, in the same stadium, Ramon Tribulietx's charges will face the Mexicans in the third-place playoff.
San Lorenzo will attempt to become the first Argentinian side to win the tournament, after Boca Juniors lost to Milan in 2007 final and Estudiantes fell to Barcelona two years later.
Auckland began the contest brightly, but San Lorenzo created the first chance, only for the unmarked Enzo Kalinski to head straight into the arms of Tamati Williams.
The Kiwis' own Argentine then threatened to hurt his compatriots. Tade, who initially moved to New Zealand to study to become a lawyer, turned Gonzalo Veron inside out but his low drive from the edge of the box presented Sebastian Torrico with a comfortable save.
Bauza's boys finally make the breakthrough two minutes into injury time. Emmanuel Mas overlapped down the left and fizzed the ball across the box. Barrientos flicked it into his own path and employed exquisite technique to volley the ball into a little gap in the bottom corner.
Within minutes of the restart, Martin Cauteruccio combined strength and skill to spin away from two defenders and race through on goal. Just as the Uruguayan was set to shoot, however, English centre-back John Irving made a superb sliding challenge.
After 65 minutes, Tade's hypnotising footwork got him past three opponents, but his cutback was hacked clear by San Lorenzo.
When he unlocked the Copa Libertadores holders' defence moments later, though, the equaliser arrived.
Tade threaded the ball between two defenders for Ryan de Vries to run onto. Somehow, the Auckland forward and the San Lorenzo goalkeeper both missed the ball, allowing Berlanga to run onto it. The Spanish defender collected it almost at the byline, but did expertly to find the back of the net from the angle.
Suddenly, San Lorenzo awoke. Cauteruccio's header from a Leandro Romagnoli free-kick forced Williams into action, before the same striker produced a sumptuous flick and volleyed against the post.
Auckland almost punished their rivals on the counter. De Vries sent Tade clean through on goal, but Torrico did well to close the distance and the attacker scuffed the ball wide.
It was onto extra time, and San Lorenzo required just three minutes of it to retake the lead. Auckland only partially cleared a long ball into their box, and substitute Matos pounced on the loose ball and fired it under Williams.
Auckland fought valiantly for a leveller, but the post denied Tim Payne's low drive and their title challenge was over. Still, it was another heroic performance from the semi-pro club, who arrived at Morocco 2014 having lost six of their eight previous Club World Cup games.
