The action came just hours after the start of a ceasefire at 10pm on Friday by Ukrainian forces, which was ordered by the country's president, Petro Poroshenko, as part of his peace plan to end the rebel insurgency in the east of the country. He urged the rebels to lay down their arms and warned Ukrainian forces would hit back if attacked.
But the rebels claimed the Ukraine had broken its own ceasefire.
A government forces spokesman said the separatists used mortars and sniper fire to attack Ukrainian posts at Izvareno and Uspenka on the border, wounding nine Ukrainian officers.
Rebels with machine guns and grenade-launchers also attacked a Ukrainian position at Avdiyivka, near Donetsk, and a Ukrainian post at Kreminna.
Separatists controlling Slaviansk also attacked Ukrainian forces on Karachun hill overlooking the town with mortars and grenade-launchers, government spokesman Vladyslav Seleznyov said.
Ukrainian forces also repelled two attacks by around 50 heavily-armed fighters early yesterday morning on an air defence base at Avdiyivka, which houses surface-to-air missiles, the defence ministry said.
The rebels have seized strategic points in major towns including Donetsk and set up "people's republics", saying they want to join Russia.
Alexander Borodai, prime minister of the self-styled Donetsk People's Republic, said the ceasefire was not working and appealed to Russia to send in peacekeeping forces.
"Since last evening, combat activities are continuing. Poroshenko's artillery is bombing Slaviansk and the air force has made several raids. Words about a ceasefire as always were just that - words," he said.
Pavel Gubarev, a prominent rebel leader, said that either Ukrainian troops were not obeying orders or Poroshenko "is lying", adding: "There is no ceasefire at all."
Poroshenko was installed as president on June 7 after seven months of turmoil in the ex-Soviet republic. He ordered government forces to cease firing to allow his 15-point peace plan to take root, with the truce due to end at 10pm on June 27.
Poroshenko is gearing up for a diplomatic push to sell his plan but his biggest challenge will be to win over Russian president Vladimir Putin. Relations with Moscow are at rock bottom and Kiev accuses Moscow of stoking the unrest.
The Kremlin on Friday denounced the ceasefire as an ultimatum rather than a peace offering and Russian foreign minister Sergei Lavrov expressed concern about Ukrainian military action.
Ukraine, for its part, expressed concern yesterday about an increase in Russian movements near the border.
In Donetsk, about 100 troops of the separatist Donetsk People's Republic took an oath of allegiance, in an apparent sign of defiance to Poroshenko's peace plan.
In a ceremony on the town's Lenin Square, armed fighters, some wearing face masks, pledged they would "defend the Donetsk People's Republic to the last drop of blood".
"We swear, we swear, we swear," they chanted in unison.