Both sides released updated death tolls as a fragile ceasefire entered its second day on Sunday.
Tajikistan said 35 of its citizens were killed, the first official death toll since armed clashes broke out on Wednesday.
On its Facebook page, the Tajik Foreign Ministry also reported 25 injured and said civilians, women and children were among the victims.
The ministry accused Kyrgyz soldiers of killing 12 people in a drone strike on a mosque, and six others in an attack on a school.
Kyrgyzstan said at least 59 people had been killed in the southern border region of Batken, and 140 injured. It said about 137,000 people were evacuated from the conflict area.
Tajikistan and Kyrgyzstan, former Soviet republics, dispute nearly 1,000 kilometres of border.
Mr Putin urged both sides to “take steps to resolve the situation as soon as possible by exclusively peaceful, political and diplomatic means”, said a statement from the Kremlin on Sunday.
The flare-up follows clashes between former Soviet states Armenia and Azerbaijan over the disputed Nagorno-Karabakh region.
Kyrgyzstan, Tajikistan and Armenia are part of the Moscow-led Collective Security Treaty Organisation, sometimes nicknamed “Russia’s mini Nato”, a group with the task of peacekeeping and security co-operation between former Soviet states.
The latest violence between Kyrgyzstan and Tajikistan is the worst since April 2021, when more than 50 were killed in clashes, and raised fears of a large-scale conflict.
The two sides agreed to a ceasefire on Friday and Mr Japarov met his Tajik counterpart Emomali Rahmon at a summit in Uzbekistan.
But the two countries traded blame for ceasefire violations only a few hours later.
Still, the situation appears to be stabilising.
On Sunday afternoon, the Kyrgyz authorities issued a statement saying the situation at the border “remains calm, trending towards stabilisation”.
“There were no recorded escalation attempts or shots fired on the border," it said.
“The parties are maintaining their ceasefire agreement and withdrawing their respective troops."
Kyrgyz President Sadyr Japarov on Monday urged his countrymen to place trust in its army and strategic partners, and said there was no need for volunteer forces at the border.
"We continue our efforts to resolve the Kyrgyz-Tajik border issues in a purely peaceful way," he said in a televised address, as the country observed a national day of mourning.
"I urge calm among the men and youths who are willing to go to Batken. We have courageous warriors and enough forces to repel those who violate our borders," he said.
On Saturday, United Nations Secretary General, Antonio Guterres, called on the leadership of both sides “to engage in dialogue for a lasting ceasefire”, a spokesman said.