Republican Governor Chris Christie may have appealed to Latinos in the Democratically-driven state of New Jersey to gain him his second gubernatorial victory, but both Democrats and Republicans are saying that doesn't mean he can win them over across the country in a 2016 presidential bid.

According to The Washington Post, some Democratic strategists said the party made a mistake by not spending more money to attack Christie during the gubernatorial campaign, which might have suppressed his margin of victory and denied the Republican a sweeping mandate in a blue state.

But Democrats shouldn't worry too much for "they plan to chip away at Christie's moderate image and present him nationally as a hard-edged conservative."

Even Christie's fellow Republicans - and potential rivals for the 2016 GOP nomination, are also saying not so fast and that Christie shouldn't get ahead of himself. Cuban-American Senators Ted Cruz and Marco Rubio downplayed his victory, Fox News Latino adds, which got him 51 percent of this crucial electorate's vote.

"I think it is terrific that he is brash, that he is outspoken, and that he won his race," Cruz, who is from Texas, told ABC News. "But I think we need more leaders in Washington with the courage to stand for principle. And in particular, Obamacare is not working."

Rubio threw in polite jab as well.

"I think we need to understand that some of these races don't apply to future races. Every race is different - it has a different set of factors - but I congratulate (Christie) on his win," he told CNN.

While both sides of the aisle may come at Christie, they have to contend with polls that lean in his favor, such as a poll conducted by Latino Decisions earlier this year. Christie ranked the third favorite of a field of possible 2016 presidential candidates, coming behind former Secretary of State Hillary Clinton, who got favorable ratings from 73 percent of the respondents, and Vice President Joe Biden, who got 58 percent.

In addition, 38 percent see Christie in a good light, surpassing San Antonio Mayor Julian Castro (34 percent) New York Gov. Andrew Cuomo (33 percent) and Senator Rubio (31 percent).

"Christie is a pragmatic politician who understands the value of listening to his constituents, even on issues his party opposes," said Cesar Vargas, director of the DREAM Action Coalition and a national activist for the DREAM Act, a measure that would allow undocumented immigrants brought to the United States as minors an chance to legalize their status. "Thus, with his support of the New Jersey Dream Act and by his proxy vote for immigration reform, it shows how a Republican can win and appeal to the Latino electorate. It's not a secret lesson but plainly obvious to everyone except House Republicans."

Colm O'Comartun, executive director of the Democratic Governors Association, points out that while Christie will be campaigning alongside Republican governors, Rick Scott of Florida, Scott Walker of Wisconsin and Tom Corbett of Pennsylvania, "he's going to have to answer the question 'Is this good for the states to focus on radical social agendas?' " Fox News Latino adds.

"What's worked for [Christie] has been to make sure that nobody talks about the issues, that people just get consumed with his personality-driven late-show entertainment," O'Comartun said. "People will see past the bluster and the vaudeville routine that is the Chris Christie show. They'll focus in on the issues."

Regardless of the feedback from both naysayers and supporters, there's no doubt Christie is relishing in his win.

"We won the Latino vote last night," he said on Wednesday at a speech in Union City, NJ. "Now find another Republican in America who's won the Latino vote recently. Why? It's because of the relationships. You get in, you build relationships, you build trust, and then people are willing to give you a chance. And of all the things that happened last night, that's the thing that I am most gratified about."