The New Year marked the end of Nato's combat operations and the start of its follow-up "training and support" mission.
About 17,000 foreign soldiers, most of them from the US, will still be deployed in Afghanistan.
Ghani, who made no mention of the wedding deaths, said Afghan forces were now solely responsible for security, as the country struggles with a worsening insurgency.
"I would like to congratulate the Afghan nation that today our security forces have become successful in defending sovereignty and taking full security responsibility," Ghani said.
"In the past 13 years, due to the problems in the region and the world, this security was a joint responsibility. Now it belongs only to Afghans. But we are not alone, we have our allies, we will continue to work together."
Officials in the southern province of Helmand said investigators were travelling to the scene to find out what caused the mortar bomb to be fired into the wedding celebrations late on Wednesday.
Many of the dead were women and children and at least 50 people were injured in the blast in Sangin district, a Taliban stronghold where US and British troops were involved in years of fierce fighting.
Karem Atal, head of the Helmand provincial council, said the Afghan army was to blame.
The Taliban issued a statement also saying the Afghan army was responsible.
"Last night, puppet security forces have fired mortars from so many directions on a wedding ceremony," they said.
"In this attack, 62 people were killed including women and children and many others wounded. We condemn this brutal and cowardly attack."
The Taliban has claimed victory with the end of Nato's combat mission, adding that no peace talks could happen before all foreign troops leave the country.
The United Nations said civilian casualties hit a new high last year in Afghanistan with about 10,000 non-combatants killed or wounded - 75 per cent of them by the Taliban.
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