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“There is an understanding and a recognition on my part that we need to reduce spending, we need to get back to balanced budgets, we need to reduce the deficit,” Mr. Donnelly said. “I hope I can be a part of making that happen.”

Sen. Tim Kaine, Virginia Democrat who won a hard-fought campaign against fellow former governor George Allen, said he is ready to get his “sleeves rolled up” and use his budgetary experience as former governor of the Old Dominion. He said the Senate’s ability to agree on a compromise bill over the New Year’s holiday “sends a good signal, but there’s much more to do.”

But Mr. McConnell said the fiscal cliff fight gives Republicans momentum and bargaining leverage in future spending-cut debates heading to the new Congress.

“President Obama declared the other night that those he calls ‘rich’ are now paying their ‘fair share,’” the Kentucky Republican said. “The president got his revenue. Now, it’s time to turn squarely to the real problem, which is spending.”