The resolution, which passed on a largely party-line vote, would fund security, enforcement and investigations at Customs and Border Protection and Immigration and Customs Enforcement, among other agencies.
It is the 13th such resolution passed by the House, all of which have stalled in the Senate. Leaders there have refused to vote on individual agency budgets, arguing instead for a budget bill that would allow the entire federal government to reopen.
"It's really not the right answer," said Rep. Ann Kirkpatrick, D-Flagstaff, of the latest resolution.
She was one of three Arizona Democrats, with Reps. Ed Pastor of Phoenix and Raul Grijalva of Tucson, to vote against the border bill. Democratic Reps. Ron Barber of Tucson and Kyrsten Sinema of Phoenix joined the state's four Republicans to support the measure.
The Congressional Budget Office estimated that the resolution would provide $19.1 billion for Customs and Border Protection, Immigration and Customs Enforcement, Coast Guard and other immigration and customs operations.
In a speech on the House floor Thursday, Barber said his first priority is "the safety and security of Southern Arizonans," and that the shutdown was causing border patrol agents "uncertainty about their pay and the future of their mission."
"These brave men and women are our first line of defense in securing the homeland," he said. "They deserve our respect and support."
Barber and Sinema were two of just 21 Democrats who broke with their party to help pass the funding measure on a 249-175 vote.
But Kirkpatrick said she hoped that instead of passing a series of spending proposals, House Republicans would allow a vote on the comprehensive spending bill.
"We could end this government shutdown tonight," she said of the comprehensive bill. "It's already gone through the Senate, the president is ready to sign it."
Rep. Paul Gosar, R-Prescott, called the "lack of cohesive voting" among Arizona Democrats interesting, and called the resolution "essential for Arizona." Individual appropriations are nothing new, he said, and the measure passed Thursday is "common sense" for the security of Arizona and the nation.
"It's sad that we have to articulate these premises to the administration," Gosar said. "The executive branch has decided that it's going to make it hurt for the citizens and individuals throughout this country if they don't get their way."
Gosar said 82 percent of the government is operating, and that number would rise to more than 90 percent if the Senate would pass the 13 funding bills already approved by the House. But that will not happen, he said, as long as President Barack Obama refuses to negotiate.
"I think everybody would start to move if the president showed some leadership in regards to not saying, ‘My way or the highway,'" Gosar said.