Luxembourg's Schleck had finished the race in second place overall, only for Spaniard Contador to be stripped of the title for failing a test for the banned anabolic agent clenbuterol and being handed a two-year retrospective ban.
Schleck was handed the yellow jersey in presence of Tour de France director Christian Prudhomme and former five-times Tour champion Bernard Hinault during a ceremony in Luxembourg.
"It's mainly about bringing the yellow jersey to the one who won it, and who would have preferred winning it in a different way and not on a green carpet, like we all would have done, but that's the way things are," said Prudhomme.
"The only thing I can repeat, and I have been saying this for a year and a half now, is the fact that the time needed to make the judicial decision and the time that rules sport, media and organisations are too far apart.
"He could have got it earlier, now finally he has, and this is fine."
Schleck is expected to take part in this year's Tour, while Contador will not be allowed to race as his suspension ends in early August.
He said: "It doesn't change anything for me, on paper I won it but on but I haven't won it on the road and that's still a goal to achieve."