About 100 cargo trucks entered the Nepali side from Sunauli border in Bhairahawa.
Home Ministry spokesman Laxmi Dhakal said Indian trucks, mostly carrying food, and six tankers loaded with oil and gas entered Nepal after a meeting between officials from both the sides.
"Although some of the trucks were released, there were thousands of trucks stranded on the Indian side in long queues," he said.
Madhesis are Indian-origin inhabitants of the Terai region bordering India who are opposed to splitting Nepal into seven provinces.
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At least 40 people have died in over a month of clashes between police and protesters from the Madhesi and Tharu communities and ethnic minorities.
Nepal government enforced the odd-even system for all kinds of vehicles plying in major cities from today in a bid to reduce the impact of the fuel crisis. It has also been rationing petroleum products in Kathmandu to tackle the problem.
Some people in Nepal blame India for the shortage by imposing an embargo.
K P Oli, chief of the ruling UML party, accused India of breaking international law by halting cross-border trade.
He asked India to release trucks carrying fuel and other essential to prove the existing friendship between Nepal and India.
"We are not getting fuel, coking gas and vegetables from India and we don't want such kind of friendship," Oli, who is tipped to be the next prime minister, said.
India denies imposing blockade, saying truck drivers are concerned for their safety after protests in the country.