"We have every confidence you're going to continue to prevail," he said.
Mr Biden's first trip to Ukraine as president came days before the first anniversary of Russia's invasion.
He said President Putin had been "dead wrong" to think Russia could outlast Ukraine and its Western allies.
He met Ukrainian President Volodymyr Zelensky and the pair visited a memorial to soldiers who have died in the nine years since Russia annexed Crimea and its proxy forces captured parts of the eastern Donbas region.
Mr Biden's presence was intended to reaffirm America's "unwavering commitment to Ukraine's democracy, sovereignty, and territorial integrity", according to a White House statement.
The New York Times reported that he took a 10-hour train journey from Poland to reach Kyiv.
He left Ukraine on Monday to travel to Poland.
After the visit, US Secretary of State Antony Blinken announced a new package of security assistance for Ukraine valued at $450m (£373m), including ammunition for howitzers and the Himars rocket system, Javelin missiles, and air surveillance radars.
The US will also provide Kyiv with an extra $10m (£8.3m) in emergency assistance "to keep Ukraine's energy infrastructure up and running", Mr Blinken said.
A new wave of sanctions against individuals and companies "that are trying to evade or backfill Russia's war machine" will also be announced later this week.
Mr Zelensky said: "The results of this visit will surely be seen and will surely be reflected on the battlefield in liberating our territories."
In his nightly address on Monday, Mr Zelensky said it was a "symbolic day" that showed "how important Ukraine is to the world".
He also said that the two leaders had discussed the possibility of sending other weapons that so far had not been supplied.
Mr Zelensky has repeatedly called for F-16 fighter jets, something the US and other allies have so far stopped short of approving.
National security adviser Jake Sullivan - who was among President Biden's entourage - said the US notified Russia about the trip a few hours before President Biden's departure for "deconfliction purposes".
He refused to comment on how Moscow had responded to the news.
White House officials said planning for the trip had been happening for "months", and that a final decision to travel was made on Friday.
In a scene that added drama to the most high-profile visit to Ukraine since the war began, air raid sirens wailed while President Biden and Mr Zelensky were in St Michael's Cathedral in central Kyiv. The sirens sound regularly in Kyiv.