The firing stopped around noon on Saturday.
"Things have been amicably settled and issues resolved. There is no tension now," said O P Gaur, chief of India's Border Security Force (BSF) at the end of a meeting between officials from both sides at Rampara border post.
"We will continue with the [construction] job we are doing but they will not. We will coordinate with them (Bangladesh) so that there is no misunderstanding."
The Bangladeshi delegation was led by Reza Sarvar, sector commander of the Bangladesh Rifles border force.
Friday's fighting began as Indian troops opened fire after Bangladesh ignored a request to stop disputed construction work on their side of the border, along a river that flows into Bangladeshi territory from India.
Heavy fighting resumed on Saturday morning, but both sides stopped firing after border authorities spoke over the telephone to fix a meeting.
The firefight took place at a border post about 350km north of Kolkata, capital of the eastern Indian state of West Bengal.
"We will continue with the [construction] job we are doing but they will not"
O P Gaur, chief,
India Border Security Force
There was no official word on casualties on either side, but Indian witnesses said at least three children had been wounded.
Indian officials said they had earlier turned down a request from Bangladeshi border authorities to allow the building of a river embankment because the issue was still being discussed by an India-Bangladesh panel on border rivers.
But Bangladesh officials denied this, saying they were trying to strengthen their river banks in line with a recent agreement between the two countries.
The fighting came a day after New Delhi offered to help Dhaka hunt for suspected Islamists who set off hundreds of crude bombs across Bangladesh, killing two people and wounding more than 100.
Normally friendly relations between India and Bangladesh have often been marred by border skirmishes, especially after New Delhi began fencing off the frontier to stop illegal immigrants and fighters.
Sixteen Indian and three Bangladeshi soldiers were killed in 2001 in the bloodiest border clash between the two sides.