MOBILE, Ala. — It took a flood of campaign donations from the business community and the backing of a large part of the Republican establishment, but Bradley Byrne, a lawyer and former state senator, successfully fought off a Tea Party-supported rival on Tuesday to become the Republican candidate for a House special election here in coastal Alabama.

The runoff was the first of what is likely to be many battles to come over the direction of the party, and it proved, to the relief of many in the Republican leadership, that a strong showing by the establishment can win tough races. But it also underscored just how difficult and costly such victories may be going forward.

Mr. Byrne opened his victory speech at a seafood restaurant here by reaching out to his opponent, Dean Young, saying he wanted to work with him to unify the Republican Party.

“If you’re in the minority, as we are in Washington, you’ve got to be unified or you can accomplish nothing,” Mr. Byrne said in a brief interview after his victory speech.