But it looked increasingly unlikely that the leader of the anti-immigrant National Front, Marine Le Pen, whose party made a shock breakthrough, coming third in Sunday's poll, would call on her followers to back him.
Hollande and Sarkozy – who beat eight other candidates – will now face each other in a run-off election on May 6.
In the first round, the Socialist beat the presidential incumbent by taking 28.63 of the vote compared to 27.18 percent for his conservative rival, according to final official results.
The first opinion poll published in the wake of Sunday's vote said Hollande would beat Sarkozy by 54 percent to 46 in the second round.
Europe's main stock markets fell sharply as traders reacted to the French election results. In Paris, the CAC 40 index slumped 2.83 percent to 3,098.37 points, its lowest level this year.
"The French poll adds to the markets' mistrust of the euro zone," said Renaud Murail, a Barclays Bourse trader.
Sarkozy moved quickly on Monday in an attempt to woo the 18 percent of voters who backed Marine Le Pen, leader of the National Front, saying they deserved an answer to their concerns, while German Chancellor Angela Merkel termed her showing "alarming".
Hollande's score cemented his position as the clear leader in the race, dealing a blow to Sarkozy's hopes of gaining enough momentum from a first-round win to defy opinion poll predictions and return to office.
But it was the showing of populist nationalist flagbearer Le Pen that shook up the race, setting up her National Front (FN) voters as potential kingmakers.
"We must respect the voters' will, it is our duty to listen," Sarkozy told journalists before heading back to the campaign trail in the Loire Valley in central France.
"There was this crisis vote that doubled from one election to another, an answer must be given."
Le Pen's score on Sunday was nearly double the 10.4 percent her father, Jean-Marie, took when he ran as FN presidential candidate in the 2007.
Hollande resumed campaigning with a trip to the western region of Brittany, where he said that the FN's score reflected anger in the country and that he too would listen to far-right voters.
"Nicolas Sarkozy is to blame for the far-right's high level," Hollande said. "There are voters who may have been led to this through anger. That is what I want to hear."
Polls show most far-right supporters prefer Sarkozy but up to a quarter – mainly working-class voters attracted by Le Pen's protectionist trade policies – could switch to Hollande.
Le Pen's high score stunned observers and she told supporters after the results that "the battle of France has just begun" and "nothing will be as it was before."
Analysts say it is extremely unlikely Le Pen will endorse either candidate in the second round, though she said she would make her views known on May 1.
"I've long considered Nicolas Sarkozy and Francois Hollande as being of a same mind on issues I consider essential, starting with the sovereignty of our country," she told French television Monday evening.
"I no longer believe in Nicolas Sarkozy's sincerity," she added.
In a speech, Sarkozy argued that French voters "tell us we no longer want a Europe that does not protect us."
"A Europe that does not regulate migration, that does not defend its borders and that opens its markets without asking for reciprocity is finished," Sarkozy said.
Sarkozy, who had already swung to the right in the campaign, brandished his right-wing credentials in his first post-results speech on Sunday.
"These anxieties, this suffering, I know them, I understand them," he said."
"They are about respecting our borders, the determined fight against job relocation, controlling immigration, putting value on work, on security," he added.
The left has not won a presidential election in a quarter of a century, but with France mired in low growth and rising joblessness, opinion polls had long predicted Hollande would beat Sarkozy.
Hollande says Sarkozy has trapped France in a downward spiral of austerity and job losses, while Sarkozy says his rival is inexperienced and weak-willed and would spark financial panic through reckless spending pledges.
Hollande has already received the backing of other left-wing first round candidates, including Jean-Luc Melenchon who took just over 11 percent of the vote and Trotskyite candidate Philippe Poutou who won 1.4 percent. Green leader Eva Joly who won 2.3 percent is also backing Hollande.