Trump boasted that "this is what winning looks like".
If the Senate voted against the bill, the government would have run out of money at midnight on Friday.
Democrats touted the lack of funding for the border wall.
Congress is expected to vote this week on the $1.1 trillion spending bill to fund the government through September.
With Democrats celebrating concessions they extracted in the spending bill, Mr Trump used a Rose Garden ceremony honouring the Air Force Academy's football team to declare he and Republicans got more from the legislation than might appear. The White House on Monday praised the deal as a win for the nation's military, health benefits for coal miners and other Trump priorities.
"The reason for the plan negotiated between the Republicans and Democrats is that we need 60 votes in the Senate which are not there!" he tweeted. He added that we "either elect more Republican Senators in 2018 or change the rules now to 51 (percent)".More news: Pakistan, Afghanistan start joint survey after border clash
That contradicted Trump's message of less than a week ago.
Democrat Rep. Eric Swalwell of California responded to Trump's call for a shutdown by suggesting that the president should be impeached. Terrible!" He also tweeted at the time that he had promised to "rebuild our military and secure our border.
Democrats quickly denounced Trump's musings about closing the federal government. But back to the current budget deal, President Trump wasn't content to let Mulvaney's briefings stand on their own.
Trump could easily trigger a partial government shutdown in October, by directing Republicans not to negotiate with Democrats or by refusing to sign a spending bill that Congress sends him for approval. He insisted that the bill will begin to fund Trump's promised border wall with money to replace fencing with steel wall.
"At the end of the day, that's who we think won in this discussion and this negotiation - not the Democrats, not the Republicans, but the American people", Mulvaney said.
Senate Majority Leader Mitch McConnell of Ky., flanked by, Sen. McConnell said there will be no more changes to the filibuster rule to make life easier for the new president.More news: Raptors beat Bucks in six games to advance
"One has to wonder if the Democrats are the majority party in Congress", said Mat Staver, the founder and chairman of Liberty Counsel, a group that advocates for conservative social policies.
Forced to spin this into something that would appease the squalling baby in the Oval Office, Mulvaney told reporters Tuesday "You can call it new wall, you can call it replacement, you can call it maintenance, call it whatever you want to". "Our country needs a White House shutdown". There are things in this bill that I just don't understand.
"While I am pleased to see increased funding for our men and women in uniform and their critical missions across the globe, this bill fails in a number of ways, including by continuing to fund sanctuary cities, Planned Parenthood, and Obamacare", Cruz said.
Any future shutdown would likely cost the federal government billions of dollars.
It was Mulvaney's third attempt in 24 hours to claim Trump did not lose, which included two conference calls, one of which was mired in technical difficulties in which he was drowned out by the song "Yankee Doodle Dandy". Patty Murray, D-Wash., a ranking member of the appropriations committee.
The White House and congressional Republicans are pressing to reverse a Washington narrative that the catchall bill is a win for Democrats. There's billions more for medical research and no changes to President Barack Obama's engagement with Cuba. He also cited $1.5 billion in emergency money for border security.More news: 'Revenge porn' on 'mass scale' in Australia
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