The win set up a semi-final clash with Colombia, who upset Ukraine 3-1, while 2008 runners-up Spain squeezed by Russia 3-2 to secure a last four match against Italy, who beat Portugal 4-3 after extra-time in the other Wednesday night tie.
Hot favourites Brazil, who are chasing a fifth world title in Bangkok, were two goals down and minutes from elimination to a determined Argentina.
But with the game slipping away, Brazil halved the deficit with a long-distance strike before talismanic forward Falcao levelled and then smashed home a winner in extra time -- a goal that saw him break down in tears.
"There are matches that go down in history," a visibly moved Falcao said after the game according to the FIFA.com website. "Today is one of those historical days, at least for me."
European champions Spain also had to come from behind to beat a dogged Russia, who went into the match boasting clean sheets in their previous four games.
Defending that record was always going to be a big ask against two-time World Cup winners Spain, whose slick interplay saw them prevail after going a goal behind.
"We won as a team," Spain coach Venancio Lopez said, of victory over the European title runners-up.
"I would like to congratulate Russia for a great match today... it's a pity that we had to face each other in the quarter-finals."
The semi-finals will be played on Friday, with the winners advancing to Sunday's final.