He was speaking just hours after President Chirac confirmed in a TV address that he would not be seeking re-election in April.
Mr Sarkozy said he had "never felt like anybody's heir".
"France is the republic, it is not something you inherit," he told French radio on Monday.
But he also praised Mr Chirac for a "dignified, sincere" speech.
Chirac under fire
Some of Mr Sarkozy's rivals in the presidential race made sharp criticisms of Mr Chirac's record during his 12 years as president.
Far-right leader Jean-Marie Le Pen said Mr Chirac had left France in a "catastrophic" state.
He will leave more popular than he has been for a long time
Le Figaro
French press bids adieu
"I think Jacques Chirac was the worst president of the republic in the history of France. With him I lose my worst enemy and I'm delighted," he said.
Socialist candidate Segolene Royal, currently second in the opinion polls behind Mr Sarkozy, called for a "new page in history" after the Chirac era.
Anti-globalisation campaigner and presidential hopeful Jose Bove dismissed Mr Chirac's record as "disastrous" and called for "a real alternative on the left that will end people's suffering and inequalities".
Centrist candidate Francois Bayrou, who has emerged as a strong contender in opinion polls, praised Mr Chirac's speech, saying it "embodied the main thread of what we should be pursuing in France".
An opinion poll published by the LH2 organisation on Monday put Mr Sarkozy on 28% in the first round, against 26% for Ms Royal and 22% for Mr Bayrou. But it also found that Mr Bayrou could beat Mr Sarkozy if he got through to the second round.
Veteran leader
Mr Chirac, 74, has been president since 1995 and has had a political career spanning four decades.
"The moment has come for me to serve you in a different way," he said in an address to the nation on Sunday evening.
Mr Chirac said he was "proud of the work which we have carried out together", citing in particular improvements for the elderly and the disabled, reforms of the pension system and reductions in crime and unemployment.
He also called on France to defend its values, saying it was a country like no other.
In a newspaper interview published on Sunday, Mr Sarkozy said he hoped to receive Mr Chirac's endorsement, but promised a new approach to politics.
In the wake of Jacques Chirac's announcement that he will not seek re-election, French politicians have taken the opportunity to distance themselves from him.

Nicolas Sarkozy, the frontrunner to succeed Jacques Chirac, said that he "never felt like anybody's heir" and that "France is the republic, it is not something you inherit."

French Prime Minister Dominique de Villepin, who was also considered a potential candidate, has endorsed Sarkozy.  Villepin and Sarkozy are members of the same party, but fell out years ago.
 
Meanwhile, right-wing leader Jean-Marie Le Pen said Mr. Chirac left France in a 'catastrophic  state.' "I think Jacques Chirac was the worst president of the republic in the history of France. With him I lose my worst enemy and I'm delighted," he said.