WASHINGTON—The Senate passed a $19.1 billion disaster aid package on Thursday, ending a monthslong impasse with an agreement that includes more assistance for Puerto Rico but no money for the southern border as sought by President Trump.

Passage came after Mr. Trump agreed to support the legislation without any border funding. The Trump administration has been seeking billions of dollars of additional moneys to address an influx of migrants from Central America, which lawmakers now plan to tackle separately.

“We’re...