Attending the Gridiron Club dinner, Mr. Obama made light of a recent back and forth between his administration and Woodward, the veteran Washington Post journalist whose reporting on the Watergate scandal helped bring down Richard Nixon's presidency.
"Can anybody tell me when an administration has ever regretted picking a fight with Bob Woodward?" Mr. Obama joked. "What's the worst that could happen?"
Mr. Woodward and White House economic adviser Gene Sperling had a public tussle that played itself out in a leaked email, in which Sperling told the veteran journalist he would regret taking a position on one of Mr. Obama's policies. Woodward suggested the move was an example of White House intimidation.
"Who knew Gene could be so intimidating," Mr. Obama joked about Sperling. "Or let me phrase it differently: who knew anybody named Gene could be so intimidating?"
The Gridiron Club and Foundation, founded in 1885, is the oldest and one of the most elite organizations of journalists in Washington. Membership is by invitation only.
Mr. Obama also ribbed Florida Senator Marco Rubio for lunging for a water bottle during his televised Republican rebuttal to Mr. Obama's State of the Union Address earlier this year.
Mr. Obama stopped his remarks briefly, deliberately picked up a water glass, sipped, then put it back down. "That, Marco Rubio, is how you take a sip of water," he said to laughter.
Mr. Rubio is a potential Republican presidential candidate in 2016.
A potential rival of Mr. Rubio's, Louisiana Governor Bobby Jindal, gave the Republican speech at the dinner, dishing out barbs about himself and other presidential hopefuls.
"I am too skinny to run," he said, referring to his chances of seeking the Republican nomination in a few years. "At least that's what my friend Chris Christie keeps telling me."
Christie, the Republican governor of New Jersey, has had admitted struggling with his weight.
Mr. Jindal said he had no plans to run for president.
"I've made that clear over and over again in Iowa, in New Hampshire and South Carolina," he said, listing states with early nominating contests that presidential contenders woo with frequent visits.
Mr. Obama, who is pressing lawmakers to end the "sequester" budget cuts that recently went into effect, joked that the often lengthy Gridiron event had not suffered from Washington's axe.
"There is one thing in Washington that didn't get cut: the length of this dinner," he said.