Shortly, he will return to the chairmanship of the House Budget Committee, where he has had a defining role in moving Republicans rightward on fiscal issues, only to confront a re-elected president intent on sticking to his campaign pledges to raise taxes on the wealthy and preserve social programs.

Mr. Ryan, whose sweeping House budget plan that sharply reduces government was embraced by Mr. Romney, can claim that he, too, won a kind of mandate. “The Ryan budget passed a pretty big test — we had to run around the country and defend it,” said Representative Tom Cole of Oklahoma, seeing in the re-election of the Republican majority in the House an endorsement for deep government cuts. The Ryan budget “proved itself a viable platform on which to run on and be re-elected.”

Mr. Ryan, who has not been a part of his party’s Congressional leadership, will become more influential in the Republican conference, Mr. Cole and others said. He will be closer to the next round of negotiations that Speaker John A. Boehner Jr. of Ohio will conduct with the White House on preventing automatic deep cuts to the military and tax increases that are scheduled to take effect at the end of the year.