"I apologise to the French people," conservative candidate Fillon told a press conference, admitting it was an "error" that he regretted "deeply".
Announcing he would not pull out of the election, due to take place in April and May, he said: "A new campaign starts this evening."
"I am the only candidate which can bring about a national recovery," he said.
Fillon, a former prime minister, called the news conference after members of his own center-right party, The Republicans, urged him to quit the race to give the party time to find a replacement candidate.
Before the scandal surfaced in a weekly satirical newspaper nearly two weeks ago, opinion polls had shown Fillon to be the clear favorite to win the election, a two-stage ballot held on April 23 and May 7, over the far-right leader Marine Le Pen.
Since then his approval ratings have plummeted and he is now seen failing to reach the May knockout.
A survey by Pollster Opinionway published on Monday showed independent centrist Emmanuel Macron, a former investment banker, challenging Le Pen in the runoff vote and winning comfortably.
The weekly Le Canard enchaine alleged that Fillon had approved hundreds of thousands of euros to be paid to his British-born wife Penelope for fraudulent work as a parliamentary assistant.
Fillon denied his wife had not properly carried out the duties of a parliamentary assistant and said the campaign against him and his wife was exceptionally virulent.
He said his wife had worked for him in several ways and represented him at functions and events.
"Her job was vital for my role as an elected official ... All of this was legal," he said.
It has been common for French legislators to employ wives, children and even mistresses in their office. But what had been acceptable practice in the past was no longer accepted by the French public, he said.
"It was a mistake, which I deeply regret, and I apologize to the French people," he said.
The stakes are high for France's Right which is battling to return to power after five years of Socialist rule under President Francois Hollande.