WASHINGTON — Senator Mitch McConnell of Kentucky decisively turned back the first well-financed Republican primary opponent he had faced since being elected in 1984, defeating a Tea Party-backed conservative who claimed the Senate minority leader had been too willing to compromise with Democrats.

Once thought to be vulnerable to such a challenge from the right, Mr. McConnell won with ease over his opponent, the businessman Matt Bevin. Mr. McConnell’s victory sets up what will be one of the most serious tests of his political career, a general election matchup against the Democratic nominee, Alison Lundergan Grimes. It is expected to be the costliest Senate race this year.