Leo's cool penalty finally sank the heartbroken visitors, who had held on for 40 minutes a man down before Kardec swooped to level proceedings
A heroic defensive effort from the Fortin kept the hosts out for almost 80 minutes, some 40 of those played with only 10 men after Marcelo Barovero saw red. Alan Kardec levelled the tie after great work from Leo and Ganso, however, meaning the lottery of the shootout decided who would enter the competition's last four.
The two teams came out to a rapturous reception in Vila Belmiro for the start of their quarter-final second leg, and the home crowd provided a deafening backdrop to events as they will their team on to overturn a one-goal deficit and book a dream semi with arch-rivals Corinthians. Barely one minute into the encounter, the Brazilians were agonisingly close to levelling the tie.
The prolific boot of Elano was almost the provider, when the former Manchester City man launched a free kick from wide on the left. The ball darted across the goalmouth, and Edu Dracena was inches away from getting the contact that would have surely led to the opener as Velez keeper Marcelo Barovero watched on.
The positive start was a misleading indication of how the first half would unfold, however. Velez naturally stayed on the back foot, looking to maintain their slim advantage and hit Santos on the break. Surprisingly the Peixe employed similar tactics in the opening stages, meaning real chances were few and far between.
Belmiro favourite Neymar was marshalled superbly by both young Gino Peruzzi and captain Fabian Cubero, and was barely given space to breathe in the opening 45. But he showed his capacity to cause damage just before half time; breaking free on the counter, the 20-year-old was chopped down by Barovero outside the box, and referee Roberto Silvera had no hesitation in sending the keeper off the field and leaving the Argentines with 10.
Elano put the resulting free kick just over the bar, while a curling effort minutes later crashed into the side netting as the teams went into half time goalless.
Muricy Ramalho's men came out far more determined to break the deadlock after the break, pressing their opponents and looking to move onto the front foot. Adriano was the first to come close, as his stinging shot from distance forced substitute keeper German Montoya into a decent save just after the restart. Velez remained resolute, however, and defended in numbers to keep their opponents away from danger.
Augusto Fernandez, meanwhile, had illusions of ending the game in the most spectacular fashion. Having forced his way past several markers, the midfielder spotted Rafael off his line and launched a brilliant lobbed effort from barely inside the Santos half; the goalkeeper scurried back and just managed to tip the ball over the bar. A storming run down the middle from Ariel Cabral then gave Augusto another chance, but from an acute angle he could only find the side-netting.
Juan Manuel Martinez also went close, and was unlucky that Augusto's cross fell on the left as he blazed wide. Muricy's frustration with his team began to show, and the coach screamed at his charges every time they failed to break the impenetrable Fortin backline. It was not just the defence who showed their worth; Montoya did brilliantly to palm away Ganso's long-range bomb to keep the scoreline in the Argentines' favour.
With 15 minutes left the Peixe faithful thought their moment had finally come. A long ball through the centre put Alan Kardec through with Velez sleeping, but the striker was frustrated by Montoya who saved his side once again. Kardec made no mistake minutes later, however, as a fluid Santos move was polished off with a left-foot finish at the edge of the box.
The late winner from Santos meant the tie went straight to penalties. Kardec, Ganso and Elano all nailed their kicks with little trouble, while Hector Canteros and Emiliano Papa missed to put Velez 3-1 down. Sebastian Dominguez had to keep his side in the tournament with his effort and beat Rafael straight down the centre, but it was to no avail as Leo sent Montoya the wrong way and sealed a heart-stopping victory.
The shoot-out triumph means Santos go through to the semi-finals of the Libertadores, keeping their hopes alive of retaining the famous trophy. The side will meet their Sao Paulo rivals Corinthians in the last four, due to the Conmebol rule that nations from the same country must play each other in the penultimate round.