Paul triumphed over Secretary of State Trey Grayson, the GOP establishment candidate, with nearly 60 percent of the vote and said opposition in the state to President Obama gives him an excellent chance in the fall.
"President Obama's less popular in our state than he's ever been. And he never was very popular in Kentucky," he said Wednesday on CBS' "The Early Show".
First up, however, is mending fences within his own party after the hotly contested primary.
"We're going to unify. I'm going to meet with Sen. [Mitch] McConnell on Saturday. We've been talking, actually, for weeks now about unifying. I've been talking with the Republican party structure, and I think we will be unified going in to the fall," Paul said.
Paul, a political newcomer, shot to prominence as a Tea Party favorite - a movement he said "is popular well outside the Republican party" as well.
Rand said he "doesn't see the Tea Party really becoming a political party," but said it would have "ramifications on both parties" and would attract independent voters.
And for his detractors who say his views are too controversial to win a general election, Paul's message is as simple as it is confident:
"I say, bring it on, and please, please bring President Obama to Kentucky. We'd love for him to campaign down here."