Late Götze strike spells heart-breaking World Cup end for Argentina
The game had seemed destined for penalties until fellow substitute Andre Schuerrle escaped down the left and sent in a cross that Goetze controlled on his chest before slamming home.
"We're going to celebrate for at least five weeks now," said Germany goalkeeper Manuel Neuer. "At some point we'll stop celebrating but we'll always keep waking up with a smile.
"We did it and it's unbelievable. In the preparation we had some setbacks. We have to think of the guys not here. They are world champions now too," he added.
Germany dominated possession but Argentina were always dangerous on the break and had the best of the few chances.
The clearest of them came after 20 minutes when Tony Kroos's misdirected header sent Gonzalo Higuain clear, only for the striker to drag his shot badly wide with the goal at his mercy.
Lionel Messi struggled to impose himself on the game and, though he shot just wide right at the start of the second half, he looked short of fitness and drifted out of the action.
As both teams tired the game went into extra time and Argentina substitute Rodrigo Palacio had a great chance when through but touched the ball wide.
Another penalty shootout had looked certain until Goetze, who began the tournament as first choice for Germany, had the final word with a memorable goal.
Having won the title in 1954, 1974 and 1990 - as West Germany - this triumph makes them the first European team to lift the trophy in the Americas.