President Petro Poroshenko ordered the Ukrainian military to hold its fire against pro-Russian militants in the nation's East. The separatist Donetsk People's Republic also announced a truce.
The ceasefire aims to end 10 months of violence in eastern Ukraine.
But less than 90 minutes after it started, Ukrainian officials reported mortar shelling at a military post near Zolote in the Luhansk region.
CNN teams in Ukraine said the shelling quieted shortly after midnight.
The "ceasefire is holding," according to chief monitor Ertugrul Apakan of the Organization for Security and Co-operation in Europe.
Referring to the first 12 hours of the ceasefire, Apakan said the ceasefire was being respected overall, with some exceptions.
Apakan noted violations in the town of Debaltseve, where an exchange of artillery fire began after the ceasefire was implemented.
Ceasefire monitors said they were blocked from traveling into the town. Deathly silence in the area was broken Sunday by the sound of shelling and at times civilians still fleeing for their lives, CNN's Nick Paton Walsh reported.
In the area around the city of Donetsk, there were reports of a mortar round, three uses of a multiple-launch rocket system and small-arms fire in the first hour of the ceasefire, but the "situation was calm," the chief monitor said.
There were also violations reported before dawn in the city of Luhansk, he said.
The OSCE mission "calls on all parties to fully adhere to the ceasefire in all locations," Apakan said.
Ukraine's Defense Ministry concurred that combat activity by "illegal armed groups" fell after the ceasefire went into effect, but said the situation in Donetsk and Luhansk regions remain "extremely difficult."
In a statement, defense spokesman Andriy Lysenko said that since the beginning of the ceasefire, "enemy forces" initiated fire from artillery, grenade launchers, mortars and small arms against Ukrainian military positions in the area near the towns of Debaltseve and Shyrokine.
Ukrainian forces had suffered no casualties since the announcement of the ceasefire, said Lysenko.
The peace agreement signed in Minsk, Belarus, comes with questions over how it will be implemented and whether it will stick. Mistrust is high on both sides, adding to the doubts.
In his speech announcing the ceasefire, Poroshenko said Ukrainian troops control the town and warned militants to stay away.
"I think the fact of using the tense situation on the outskirts of Debaltseve by militants-terrorists-mercenaries poses a great threat, which can violate the ceasefire regime," Poroshenko said.
Separatists claimed to have encircled Debaltseve.
Poroshenko has warned that if the separatists do not abide by the ceasefire, he will impose martial law throughout Ukraine's territory.
Shelling hours before
Shelling rocked two eastern Ukrainian cities Saturday ahead of the ceasefire, raising fears that the deal to end the bitter conflict might be in jeopardy.
Both incoming and outgoing artillery could be seen in the vicinity of the coastal city of Mariupol, and there was significant shelling in the city of Donetsk.
Ukraine's National Defense and Security Council reported seven soldiers killed and 23 wounded in 24 hours. The council said additional forces were crossing the border from Russia into Ukraine in the area of Novoazovsk.
Igor Martinov, separatist-appointed mayor of Donetsk, said two people were killed and four injured by shelling in the city center Saturday afternoon.
OSCE urges reduction of hostilities
Under the deal, both sides were expected to start pulling back their heavy weapons from the front lines Monday, creating a buffer zone at least 50 kilometers (31 miles) wide.
The OSCE has the challenging task of overseeing the process.
It also envisions prisoner swaps, new local elections in the conflict-affected regions of Luhansk and Donetsk, and a law granting amnesty to those involved in the fighting.
However, Svyatoslav Tsegolko, a spokesman for the Ukrainian president, said that those responsible for shooting down Malaysia Airlines Flight 17 over Ukraine last summer will not be given amnesty under the Minsk agreement.
This could be a potential obstacle to keeping the separatists on board.
Western governments believe rebels used a Russian-supplied missile system to shoot down the jet after mistaking it for a military aircraft, while Kiev says it was a Russian officer who pressed the button. The separatists and Russia have denied any responsibility.
Russia 'not a participant' in conflict
The Kremlin has said it hopes to see the latest peace plan succeed, but reiterated Russia "is not a participant" in the conflict.
Russia has steadfastly denied accusations by Kiev and the West that it is sending forces and heavy weapons into Ukraine.
But Kiev has accused it of continuing to do so even after the peace plan was signed in Minsk.
A previous peace agreement in September swiftly disintegrated.