Mr Biden warned that Moscow has no intention of ending its "cruel war" and vowed American support for Kyiv for "as long as it takes".
Wearing his signature green combat fatigues, the president thanked Mr Biden, the US Congress and ordinary Americans for financing Ukraine's war effort.
"I came here to the United States to ... thank the people of America, people who do so much for Ukraine. I am thankful for all of this," Mr Zelenskyy said.
After Mr Zelenskyy's visit to the White House, he went to Capitol Hill for an address to Congress, where he petitioned for increased support as Republicans prepare to take over the House of Representatives next year.
He was largely met with visible enthusiasm and was interrupted by standing ovations throughout much of his speech.
Some Democratic members of Congress greeted Mr Zelenskyy by waving a large Ukrainian flag in the chamber.
"Ukraine holds its lines and we will never surrender," he said.
Congress was due to vote on a spending package that includes about $45 billion in emergency assistance, which comes on top of military and economic donations throughout 2022 that have totalled about $48 billion, far more than any other country has given during the 10 months of conflict.
The war's financial cost has become an increasing cause for concern among US taxpayers, particularly Republicans, and Mr Zelenskyy's visit was designed to shore up support for continuing spending.
Republican scepticism over support for Ukraine was apparent on corners of the chamber floor.
Far-right representatives Lauren Boebert and Matt Gaetz refused to applaud and were seen laughing during Mr Zelenskyy's address.
As the tides prepare to turn in the US Congress, the Ukrainian President said support for Ukraine protected freedom around the world.
"Your money is not charity. It is an investment in the global security and democracy that we handle in the most responsible way," Mr Zelenskyy said.
He also called out the use of Iranian-made drones in Russia's assault on Ukraine.
"It is just a matter of time when they will strike against your other allies," he said.
Mr Zelenskyy's trip coincided with the announcement of the biggest US arms delivery to Ukraine to date, including Patriot surface-to-air missiles.
"The United States is committed to ensuring that the brave Ukrainian people can continue continue to defend their country against Russian aggression as long as it takes," Mr Biden said.
The US leader said Russian President Vladimir Putin, who has ordered the destruction of civilian infrastructure and the energy grid, had no intention of stopping his "cruel war" and described how Russia was using winter as a weapon to freeze and starve Ukraine into submission.
"It's the latest example of the outrageous atrocities the Russian forces are committing against innocent Ukrainian civilians, children and their families," Mr Biden said.
“For me as a president, ‘just peace’ is no compromises,” Mr Zelenskyy told reporters.
He said many Ukrainians are living now "for the sake of vengeance" and said the war could only end once Ukraine’s sovereignty, freedom and territorial integrity were restored, as well as “payback for all the damages inflicted by Russian aggression.”
Mr Zelenskyy praised Washington's earlier announcement that it would be including the first transfer of the Patriot air defence system as part of a $1.85 billion military aid package.
The air defence system is capable of shooting down Russian missiles and enemy aircraft.
"This is a very important step to create a secure airspace for Ukraine and that's the only way we would be able to deprive the terrorist country," Mr Zelenskyy said.
As part of his visit, he must also convince some Republicans in Congress to continue backing Ukraine with funds.
The party's top politicians in the House have suggested they would pare back funding when they took control next month.
"I want to win together," said Mr Zelenskyy.
At a meeting inside the Oval Office earlier on Wednesday, Mr Zelenskyy presented Mr Biden with a medal earned by a Ukrainian missile captain who wanted to give the award to the US President.
"He's very brave, and he said, 'Give it to a very brave president'," Mr Zelenskyy said, recounting how he had met the leader of a rocket unit in the eastern city of Bakhmut on Monday.
The trip came on the same day that Mr Putin announced Moscow would continue to develop its military and the readiness of nuclear forces.
The Kremlin said the Patriot systems offered to Ukraine by the US would only aggravate the war, but Mr Biden said it is not an escalatory action as the missiles are defensive.
Mr Zelenskyy's Washington visit comes on the 300th day of a war in which Russian forces have been halted by a stubborn Ukrainian army with western arms.
Almost 10 months after invading Ukraine, Moscow's troops have been forced to retreat from captured territory in the past three months, including the former frontline city of Bakhmut.
Mr Zelenskyy was invited to address Congress by House Speaker Nancy Pelosi, who said his leadership had “rallied not only your people, but the world, to join the front lines of the fight for freedom”.
The visit was quietly planned, beginning with a phone call between Mr Biden and Mr Zelenskyy on December 11, followed by a formal invitation a week ago and confirmation of the visit on Sunday.
Meanwhile, the US Senate advanced a government funding bill that includes $44.9 billion in emergency assistance to Ukraine and Nato allies.
The money will be used for military training, equipment, logistics and intelligence support, and for replenishing US equipment sent to Ukraine.