"We fought the good fight; we just didn't win," Boehner told a radio station in his home state of Ohio.
Democratic Sen. Chuck Schumer of New York blasted Cruz and the rest of the tea party wing in Congress for what he called the "reckless, irresponsible politics of brinksmanship over the last few weeks."
"It was not America's finest moment," he said.
Markets soar on agreement
News of the deal brought some relief to Wall Street as well as Washington, with pressure to resolve the impasse building with the approach of the Thursday deadline to raise the debt ceiling or face default.
U.S. stocks rose on the news of an agreement, with the benchmark Dow Jones Industrial Average jumping more than 200 points on the day.
Reid hailed the agreement he worked out with McConnell as "historic," saying that "in the end, political adversaries put aside their differences."
McConnell fired an opening salvo for the budget talks expected to begin soon and continue until December when he said any ensuing spending deal should adhere to caps set in a 2011 law that included forced cuts known as sequestration.
"Preserving this law is critically important to the future of our country," McConnell said of the Budget Control Act, which resulted from the previous debt ceiling crisis in Washington.
The focus on an agreement shifted to the Senate after House Republicans failed on Tuesday to come up with a plan their majority could support, stymied again by demands from tea party conservatives for outcomes unacceptable to Obama and Senate Democrats, as well as some fellow Republicans.
Cruz, despite being in the Senate, is credited with spearheading the House Republican effort to attach amendments that would dismantle or defund the health care reforms known as Obamacare to previous proposals intended to end the shutdown.
All were rejected by the Democratic-led Senate, and Obama also pledged to veto them, meaning there was no chance they ever would have succeeded.
Republican Sen. Kelly Ayotte of New Hampshire called the House GOP tactic of tying Obamacare to the shutdown legislation "an ill-conceived strategy from the beginning, not a winning strategy."