Obama dismissed the charge out of hand, then turned the jeers to applause when he countered, "What we shouldn't be doing is tearing each other down, we should be lifting the country up."
The exchange marked an unusually pointed moment in an otherwise civil encounter in the days before March 4 primaries in Texas and Ohio - contests that even some of Clinton's supporters say she must win to sustain her campaign for the White House.
The former first lady has lost 11 straight primaries and caucuses, and trails her rival in convention delegates. Obama has won a pair of big union endorsements in the past two days.
Overall, the night did little to stop Obama's momentum, which has helped him pull even with Clinton in Texas polls after trailing her by double-digits only a few weeks ago, CBSNews.com senior political editor Vaughn Ververs said.
"Anyone looking for a knock-down, drag-out debate in Texas tonight was instead treated to a gentle waltz that only increased Obama's edge over Clinton," Ververs said. "The New York Senator did nothing to stop Obama's momentum and, in fact, allowed him to upstage her on both substance and style. Time is growing short for Clinton to regain her footing. This debate just kept those seconds rapidly moving forward. And the time is all on Obama's side."
In a university auditorium in the heart of Texas, the two rivals agreed that high-tech surveillance measures are preferable to construction of a fence to curtail illegal immigration.
They disagreed on the proper response to a change in government in Cuba in the wake of Fidel Castro's resignation. Clinton said she would refuse to sit down with incoming President Raul Castro until he implements political and economic reforms. Obama said he would meet "without preconditions," but added the U.S. agenda for such a session would include human rights in the Communist island nation.
They also sparred frequently about health care, a core issue of the campaign.
Clinton said repeatedly that Obama's plan would leave 15 million Americans uncovered.
But he, in turn, accused the former first lady of mishandling the issue by working in secrecy when her husband was in the White House.
"I'm going to do things differently," he said. "We can have great plans, but if we don't change how the politics is working in Washington, then neither of our plans are going to happen."
Clinton was combative and complimentary by turns, and reflected on her well-known personal struggles in the debate's final moments.
"Everyone here knows I've lived through some crises and some challenging moments in my life," she said - a thinly veiled but clear reference to her husband's affair with Monica Lewinsky and subsequent impeachment. But she added that nothing she had been through matched the everyday struggles of voters.
Then, offering unprompted praise to her rival, the one-time front-runner said, "No matter what happens in this contest, I am honored to be here with Barack Obama."
Both candidates were plainly popular with the debate audience. During one break someone in the crowd shouted "Si, se puede," Spanish for Obama's trademark phrase, "Yes we can."
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