Following the Tuesday special election win of a Democrat over a Republican for a vacant House seat in upstate New York - a seat, typically held by Republicans - former President Bill Clinton had some steady advice for his fellow Democrats: Don't get too carried away.

The win was considered by many to be a major coup against the Medicare overhaul proposed by Rep. Paul Ryan, R-Wis. -- a plan that would phase out Medicare as a fee-for-service system and replace it with a private insurance program backed by government subsidies. People 55 and older would not be affected by the plan.

Clinton, speaking Wednesday at a fiscal summit sponsored by the Peter G. Peterson Foundation in Washington, said the race proved voters don't like the Republican plan, but that doesn't mean, he said, that Democrats should not continue to focus on entitlement programs, CNN reported.

You shouldn't look at the New York race and think that nobody can do anything to slow Medicare costs, he said.

Clinton's tempered words come in seeming contrast to those of his Democratic counterparts.

After the win of Democrat Kathy Hochul over Republican candidate Jane Corwin for the vacant seat in New York's 26thCongressional District, House Minority Leader Nancy Pelosi of California issued a statement, saying, victory in a staunchly Republican district has shocked the political world and sent an unmistakable sign that the American people will not stand for the Republicans' reckless and extreme agenda to end Medicare, the San Francisco Chronicle reported.

And these words by Democratic Congressional Campaign Committee Chairman Steve Israel, D-N.Y.: The three reasons a Democrat was elected to Congress in the district were Medicare, Medicare and Medicare, he said in an interview with the Washington Post.

Speaking after Clinton at the fiscal summit, Rep. Ryan repeated his earlier response to the Democratic win by saying, Democrats are shamelessly demagoguing and distorting this. Trying to scare seniors and making these things as political weapons creates political paralysis, he said.

He also said that Hochul's campaign was built on scare tactics. To which, Tevi Troy, a senior fellow and consultant on health care and domestic policy at the Hudson Institute, agreed.

He is right; it was a 'Mediscare' campaign, Troy said. People see it as a political opportunity to throw stones at [the Medicare proposal]. I applaud him for coming up with a plan for trying to get us out of this mess.

Troy also pointed out that the GOP has a tendency to do poorly in special elections that are focused narrowly on local issues.

The GOP does better on broader, national issues, Troy said. If we had had a national race going on, not a special election, I think the outcome would have been different. But I don't see the seat staying Democrat in the long haul, he said.