Mr Obama said Mr Romney wanted to return to the policies that hurt the US economy.
Mr Romney said Mr Obama's policies had failed and America needs a change. "Going forward with the status quo is not going to cut it for the American people who are struggling today," Mr Romney said.
The showdown was critical for Mr Romney, a former Massachusetts governor. With five weeks to go before the election, polls show Mr Obama leading in the most important states in what is a state-by-state vote that decides the presidency.
Still, the race remains tight and the three debates this month give Mr Romney an opportunity to shift the momentum, taking on Mr Obama before a television audience of tens of millions.
This debate might be the most important of the three, with its focus on domestic issues. The weak economy, with unemployment at 8.1%, has dominated the race.
Mr Romney was particularly aggressive, like a man looking to shake up the campaign with a little less than five weeks to run. He said that under Mr Obama's policies "middle-income families are being crushed".
Mr Obama has argued that he prevented a meltdown after inheriting an economy in free fall from the Republican administration of George W. Bush.
Mr Obama accused Mr Romney of wanting to "double down on the top-down policies" that led to the economic crash four years ago.
After Mr Romney said he would repeal and replace regulations passed after the 2008 financial crisis, Mr Obama responded: "Does anyone think there is too much oversight and regulation of Wall Street?" If so, "Gov. Romney is your candidate".
Mr Obama said his opponent's plan to reduce all tax rates by 20% would cost five trillion US dollars and benefit the wealthy at the expense of middle income taxpayers.
Mr Romney shot back: "Virtually everything he just said about my tax plan is inaccurate."
But the debate began on a friendly note. The two rivals clasped hands and smiled as they strode onto the debate stage at the University of Denver, then waved to the audience before taking their places behind identical podiums. They faced questions from moderator Jim Lehrer of PBS.
There was a quick moment of laughter, when Mr Obama referred to first lady Michelle Obama as "sweetie" and noted it was their 20th anniversary.
Mr Romney added best wishes, and said to the first couple: "I'm sure this is the most romantic place you could imagine, here with me."
Though Election Day is more than a month away, many Americans have already started casting ballots because some states allow early voting. That puts extra pressure on Mr Romney to come up with a showing strong enough to alter the course of the campaign.
Mr Romney took part in 19 debates during the campaign for the Republican primary early in the year. Mr Obama has not been onstage with a political opponent since his last face-to-face encounter with John McCain, his Republican rival in 2008.
The next two debates are on October 16 in New York and on October 22 in Florida.
Vice President Joe Biden and Mr Romney's running mate, congressman Paul Ryan, have one debate, on October 11 in Kentucky.