WASHINGTON—The Senate's top Democrat and Republican opened negotiations on Saturday aimed at avoiding a U.S. debt crisis and reopening the government, marking a new chapter and new urgency in efforts to resolve the political stalemate in Congress.

The talks between Senate Majority Leader Harry Reid (D., Nev.) and Minority Leader Mitch McConnell (R., Ky.) were their first face-to-face negotiations since the government shutdown began on Oct. 1, and showed a changed dynamic in the Capitol. House Republicans, a dominant force in...