The annual tradition has the president mocking others and himself as celebrities mingle with journalists and politicians.
Mr Obama made fun of conservative television hosts' talk about Mr Putin's bare chest, and he joked about opposition claims that he had been born overseas instead of the US.
Looking ahead to a possible successor in office, he said it will be a lot harder for Republicans to prove that Hillary Rodham Clinton was born in Kenya.
The president also poked fun at CNN's extensive coverage of the search for the missing Malaysia Airlines plane.
He said he was still jet-lagged from his visit in recent days to Malaysia and added: "The lengths we have to go to get CNN coverage these days!"
The president also turned the humour on himself and his difficult 2013, including the deeply troubled roll-out of the website for his health care overhaul.
First Lady Michelle Obama accompanied him to the black tie event, and Joel McHale, star of the NBC series Community, was the dinner's featured entertainer.
Celebrities attending included Sofia Vergara and the stars of Washington-centric shows like Veep and Scandal.
The dinner has often come at key moments of Mr Obama's presidency.
In 2011, he showed up the day before special operations troops killed Osama bin Laden. Last year's dinner came nearly two weeks after the deadly Boston Marathon.
This time, the US and Europe are anxiously watching Ukraine and Russia's role in the turbulence in the eastern region of the former Soviet state.
The correspondent's association, which represents the White House press corps, is celebrating its 100th anniversary.