The International Committee of the Red Cross, which had planned to escort the convoy to assuage fears that it was being used as a cover for an invasion by Russia, said it had not received enough security guarantees to escort the convoy.
Trucks loaded with water, generators and sleeping bags sent from Moscow are intended for civilians in the city of Luhansk, where pro-Russian separatist fighters are besieged by government forces. Shelling of the city has been ongoing for weeks.
An Associated Press reporter saw a priest blessing the first truck in the convoy at the rebel-held checkpoint and then climbing into the passenger seat. A rebel commander on the scene said 34 trucks had gone through. On the Russia side of the border, an Associated Press reporter counted another 32 vehicles going into the customs zone.
The vehicles' immediate destination was not known and it was not clear whether Kiev had granted its approval.
"The Russian side has decided to act," said a statement on the Russian foreign ministry's website. "Our column with humanitarian aid is starting to move in the direction of Luhansk."
The Red Cross said in a statement on its Twitter account that it is not escorting the convoy due to security concerns, as shelling had continued overnight. "We've not received sufficient security guarantees from the fighting parties," it said.
A rebel commander on the scene who identified himself only by the codename Kot said the trucks were headed for the city of Luhansk.
Shortly after leaving from the border town of Izvaryne, the convoy departed from the main road to Luhansk and headed north onto a country road and parked in the village of Uralo-Kavkaz. That route also leads to Luhansk, potentially avoiding areas controlled by government troops.
The relief supply mission is proceeding despite both Ukraine and the separatists ignoring pleas for a cease-fire.
The trucks had been stranded in a customs zone for more than a week since reaching the border, as the Russian foreign ministry voiced increasing frustration at what it said were Kiev's efforts to stall its delivery.
A total of 34 Russian vehicles had received initial approved from Ukraine on Thursday but were awaiting a final green light. But Russia's foreign ministry blasted the Ukrainian side for "endless, concocted delays" in delivering the aid, which left Moscow early last week.
It blamed the Kiev government for continuing to shell residential areas that the convoy would have to pass through, thereby making its onward travel impossible.
"There is increasingly a sense that the Ukrainian leaders are deliberately dragging out the delivery of the humanitarian load until there is a situation in which there will no longer be anyone left to help."
Laura Mills in Moscow contributed to this report.