The temporary measure, pushed through to avert a government shutdown, did not include $6bn (£4.92bn) in military aid for Kyiv - a top White House priority.
Hardline Republicans oppose further military aid, with many openly opposing Mr Biden's approach to the war.
But on Sunday Mr Biden said Ukraine could "count on" US support.
"We cannot, under any circumstances, allow US support to Ukraine to be interrupted," Mr Biden said.
"I can reassure [Ukraine] we'll get there, that we're going to get it done," he said on restoring funding for the war. "I want to assure our American allies... that you can count on our support, we will not walk away."
The US has already supplied some $46bn (£37bn) in military aid to Ukraine since Russian launched its full scale invasion in February 2022.
President Biden has requested another $24bn (£19bn).
And in recent months the US has sent state of the art equipment to Kyiv - including long-range missiles and Abrams tanks. It comes as Kyiv's forces continue to launch a slow moving counter-offensive in the south of the country.
But Saturday's temporary budget agreement - which will fund the US federal government for 45 days - stripped out continued military funding for the time being.
Senior Senate leaders from both parties released a joint statement signalling their intention to "ensure the US government continues to provide" support to Ukraine in the coming weeks.
But the move - which came just nine days after President Volodymyr Zelensky flew to Washington to plead for further support - reflects increasing opposition from hard-right Republicans in the House of Representatives to the war in recent months.
Republicans control the House of Representatives, with Democrats enjoying a wafer-thin majority in the Senate. Both need to approve legislation on the budget before it is signed into law.
