Spokesman Jay Carney said Obama spoke with six foreign leaders as he continues to return messages of congratulations from heads of state around the world.
Carney said Obama thanked each leader for his friendship and expressed a desire to continue cooperation in the future.
The Kremlin said Putin repeated an invitation to Obama to visit Russia.
A statement from Karzai's office said Karzai had accepted an invitation to visit the United States for talks.
The White House said Obama also spoke with the leaders of Italy, Jordan, Qatar, and Spain.