"I have a message, a message from the Tea Party, [a] message that is loud and clear and does not mince words. We have come to take our government back," said Paul.

David Hawkings said the victory shows that the Tea Party is a force to be reckoned with.

"It is a clear sign that the Tea Party movement has some lasting staying power, that is has an ability to galvanize disaffected voters, and not only did Rand Paul win in Kentucky, but a Tea Party candidate won in a House race in Louisville, [Kentucky] defeating another establishment candidate," said Hawkings.

But Hawkings said the Tea Party victories could actually turn out to be a disadvantage for Republicans in the November general elections, because Tea Party candidates tend to be viewed as too far to the right ideologically by moderate voters.

"It is an age-old problem in politics, where in general in American politics, the candidate who appeals most to the most fervent members of the party wins the primary and then often has a very difficult time appealing to the broader electorate," added Hawkings.

The party that controls the White House historically loses congressional seats in a new president's first mid-term elections. The question for President Obama and Democrats is how many seats they will lose. Republicans would need a 40-seat gain in November to take back majority control in the U.S. House of Representatives.

Hawkings and other analysts caution political observers not to read too much into Tuesday's results, because there are still more than five months to go until the November elections.