Obama scolded his challenger for being "all over the map" on foreign policy but Romney appeared to have passed the "commander-in-chief" test of looking authoritative on national security issues.
With two weeks left until election day, the high-stakes debate strayed frequently into domestic policy, with Romney seeking to bolster his argument that Obama had bungled the US economic recovery.
Running neck and neck in polls, neither man threw a knockout punch or made a noticeable gaffe as they clashed over Israel, Iran, Russia and the size of the US Navy in the encounter at Lynn University in Boca Raton.
While tamer than the second debate last week in New York state, the matchup had its share of zingers and putdowns, most of them doled out by an aggressive president eager to stop a surge in polls by the former Massachusetts governor.
"I know you haven't been in a position to actually execute foreign policy, but every time you've offered an opinion, you've been wrong," said Obama.
"Attacking me is not an agenda," was Romney's frequent retort, alluding to Republican accusations that Obama had not laid out enough of a policy plan for a second term.
Snap polls declared Obama the winner, but 60% of people in a CNN survey said Romney was capable of being commander in chief, accomplishing a key goal set out by his advisers.
A CBS News poll said 53% believed Obama won the debate, versus 23% for Romney and 24% calling it a draw. The CNN poll put Obama as the winner by 8 percentage points.
With foreign policy a low priority in a campaign focused on the economy, it was unclear what impact the debate would have on the race. Respondents in the CNN poll were split over whether it would influence their votes in the November 6 election.
The campaign now enters its decisive phase with two weeks of campaign rallies across battleground states. Polls show a tied race, after Romney clawed back from a deficit by outdueling Obama in their first debate on October 3.
The Boca Raton showdown was one last chance for the candidates to appeal to millions of voters watching on television and Obama was the aggressor from start to finish.
He criticised the Republican for lacking ideas on the Middle East, mocked his calls for more ships in the US military and accused Romney of wanting to bring the US back to a long-abandoned Cold War stance.
Horses and bayonets
"On a whole range of issues, whether it's the Middle East, whether it's Afghanistan, whether it's Iraq, whether it's now Iran, you've been all over the map," Obama said.
He had a biting and perhaps condescending response when the Republican said the US Navy had fewer ships now than at any time since 1917 and needed more.
"Governor, we also have fewer horses and bayonets," Obama said, suggesting that Romney's worldview was obsolete. "We have these things called aircraft carriers, where planes land on them. We have these ships that go underwater, nuclear submarines."
Romney was cautious throughout and often refused to take the bait when attacked, which may have led viewers to declare Obama the winner.
The former business executive's approach seemed to be a sign that he feels he has the momentum in the campaign and was trying to avoid the kind of mistakes he made on a trip abroad in July to London, Jerusalem and Poland. He often steered the conversation back to the economy.
"The president's path means 20 million people out of work struggling for a good job," Romney said in his closing statement, delivered as he sat at a table with Obama and facing CBS News' moderator Bob Schieffer. "I'll get people back to work with 12 million new jobs."
Romney accused Obama of failing ally Israel, which the Democrat has not visited since taking office four years ago.
Both men declared, however, they would defend the Jewish state if it were attacked by Iran and both vowed to pursue tough policies against Tehran's nuclear ambitions and keep military action as a last resort. Iran says its atomic program is for peaceful purposes.
Increasing the pressure, Obama said the Republican presidential candidate was seeking to turn back the clock to the 1980s by once declaring Russia the top "geopolitical foe" of the US.
The 1980s, said Obama, were "now calling to ask for their foreign policy back because, you know, the Cold War's been over for 20 years."
Romney pulled his punches on Libya, a potential weak spot for Obama after the White House gave a shifting story about the September 11 attack by militants on the US Consulate in Benghazi in which four Americans were killed.
But he did concentrate on Obama's handling of crises throughout the Middle East, from Syria to Egypt and north Mali.
Romney said Obama's policies toward the Middle East and North Africa were not stopping a resurgence of the threat from al Qaeda in the region.
While Obama has conducted a policy of using unmanned drones to attack al Qaeda targets, Romney said more was needed, a "comprehensive and robust strategy" to persuade the world to reject militant Jihadists.
"We can't kill our way out of this mess," he said.
On China, both vowed to get tough on Beijing's trade policies, but touched only briefly on the security challenge the fast-rising country poses to a US-led alliance system in Asia.
Obama suggested he would not allow drastic reductions in defence spending as part of $1.2-trillion in proposed government cuts over 10 years, known as "sequestration." The cuts would begin taking effect if Obama and Congress cannot reach a deficit-reduction deal by January 2.
"First of all, the sequester is not something that I've proposed. It is something that Congress has proposed. It will not happen," he said.
Advisers on both sides were quick to declare that their men did what they had to do.
Obama deputy campaign manager Stephanie Cutter said Obama may have come across as aggressive "because Mitt Romney could not go beyond his talking points."
Romney's aides said he did enough to convince voters he was knowledgeable about foreign affairs and could handle the responsibilities of the Oval Office.
"President Obama is falling behind in the race for president. So he shows up and launches one attack after another. He looked desperate," said Romney senior adviser Eric Fehrnstrom. – Reuters