Your memory is very incorrect. In Clinton's Address Before a Joint Session of Congress, he said that he wanted to raise taxes and cut defense spending (aka, right wing welfare).
http://www.presidency.ucsb.edu/ws/?pid=47232
Newt Gingrich then rallied+campaigned against this and the Omnibus Budget Reconciliation Act of 1993 received zero Republican votes. The fiscal conservatism was all Clinton; but the conservatism was all Republican.
https://en.wikipedia.org/wiki/Omnibus_Budget_Reconciliation_...
Gingrich went on to campaign against this in the 1994 midterms. He went on to shut down the government twice to say nothing of the impeachment nonsense while he was cheating on his own wife. Gingrich was booted out by his own majority and was reprimanded by them for his own (lack of) ethics.
No, Republicans get zero, nada, no credit for Clinton's budgets and his ensuing surpluses.
As I recall, Clinton cooperated with a fiscally-conservative Republican Congress to get the budget in line. Rapid growth in the IT industry helped raise revenues as well. Unfortunately the Republicans turned against him at the end, and completely lost their fiscally-conservative policy standpoint. Deficit spending has just been getting worse and worse since then, regardless of who's been in power.
http://federal-budget.insidegov.com/ [1]