Popular Party leader Mariano Rajoy has told supporters that Spain's voice must be respected again.
At the end of a campaign dominated by the debt crisis, the PP secured 186 of the 350 seats in the lower house of parliament.
However, Spain's borrowing costs continued to mount on Monday and the stock market fell by 3.14%.
"There won't be any miracles. We never promised any," said Mr Rajoy.
Speaking outside party headquarters in central Madrid, he assured his audience:
"We will stop being part of the problem and will be part of the solution."
The PP won 44.62% of the vote in a turnout of 71% of the electorate.
The Socialists, who have been in power since 2004, slumped to 110 seats with 28.73% of the vote.
Outgoing Prime Minister Jose Luis Rodriguez Zapatero said he would co-operate with his successor for the good of the country.
"The most important thing at this moment is that Spain stays solvent," he said.
PP campaign co-ordinator Miguel Arias said Spain was "going to make all the sacrifices".
"We have been living as a very rich country," he told BBC News.
But the BBC reports Mr Rajoy, 56, has so far provided little indication of how he aims to calm the markets and fight Spain's economic crisis.
He is not due to take office until 22 December.
Markets down
Markets in Spain fell on Monday despite the resounding victory for the Popular Party on Sunday.
The Ibex 35 stock index fell 3.14% in afternoon trading and the government's cost of borrowing rose.
On Monday, the yield on the 10-year bond rose a fifth of a percentage point to 6.49%.
Spain's borrowing costs on Thursday reached their highest level since the eurozone was formed, when 10-year bonds were sold at an interest rate of 6.975% at an auction.
The unemployment rate in Spain is 21.5%, its highest for 15 years and more than twice the European Union average.
Total unemployment is five million and the jobless rate for non-students aged under 25 is 46%.
