"It's about creating a pathway forward that leaves Ukraine sovereign, independent and prosperous, and so we expect to make even more progress today," Rubio said in Hallandale Beach, Florida, where the meeting took place.
Sunday's discussions follow roughly two weeks of negotiations that began with a US blueprint for peace that critics said initially favoured Russia, which started the Ukraine conflict with a 2022 invasion.
US President Donald Trump has expressed frustration at not being able to end the war.
He pledged as a presidential candidate to do so in one day and has said he was surprised it has been so hard, given what he calls a strong relationship with Russian President Vladimir Putin, who has largely resisted concessions to stop the fighting.
Trump's team has pressured Ukraine to make significant concessions itself, including giving up territory to Russia.
The talks shifted on Sunday with a change in leadership from the Ukrainian side.
A new chief negotiator, national security council secretary Rustem Umerov, led the talks for Ukraine after the resignation on Friday of previous team leader Andriy Yermak, chief of staff to Ukrainian President Volodymyr Zelenskiy, amid a corruption scandal at home.
As the meeting began, Umerov thanked the United States and its officials for their support.
"US is hearing us, US is supporting us, US is walking besides us," Umerov said in English.
The talks got off to a "good start" and were taking place in a "warm atmosphere," Ukrainian first deputy foreign minister Sergiy Kyslytsya said on Sunday.
"It's been a good start of the ongoing meeting. Very engaging and so far constructive. Warm atmosphere, conducive to potential progressive outcome," he wrote on X.
Special envoy Steve Witkoff and Trump's son-in-law Jared Kushner were also present to represent the US side.
Witkoff is expected to meet Russian counterparts later this week.
The talks took place near Miami at a private club, Shell Bay, developed by Witkoff's real estate business.
Zelenskiy had said he expects the results of previous meetings in Geneva would be "hammered out" on Sunday.
In Geneva, Ukraine presented a counter-offer to proposals laid out by US Secretary of the Army Dan Driscoll to leaders in Kyiv two weeks ago.
Ukraine's leadership, facing a domestic political crisis fuelled by a probe into major graft in the energy sector, is seeking to push back on Kremlin-friendly terms as Russian forces grind forward along the front lines of the war.
Last week, Zelenskiy warned Ukrainians, who are weathering widespread blackouts from Russian air strikes on the energy system, that his country was at its most difficult moment yet but pledged not to make a bad deal.
Turkish Foreign Minister Hakan Fidan said Russia and Ukraine are increasingly willing to reach a peace deal in remarks to the Germany's Welt am Sonntag newspaper.
"After four years of attritional war, the parties are more prepared to conclude peace than previously," Fidan said.
"They have seen the extent of human suffering and the destruction, and recognised their own limits," he added.
Turkey's understanding was that Putin was now prepared to agree to a peace deal under certain conditions, and this attitude had been conveyed to the Ukrainian side, Fidan said.
Turkey was involved "in certain aspects of this," he added, noting that the war was exacting a heavy toll on Russia as well.
with DPA
