0330 GMT November 16, 2018
Hundreds of people from the two countries hugged each other and some wept as their leaders led celebrations to mark the reopening, BBC reported.
It is the latest sign of rapprochement between the former enemies.
Ethiopian Prime Minister Abiy Ahmed and Eritrean President Isaias Afwerki signed a peace deal in July.
It restored diplomatic and trade relations between the nations.
The reopening at Burre gives landlocked Ethiopia access to the sea. Another border post, near the Ethiopian town of Zalambessa, also reopened.
The reopening coincides with the Ethiopian New Year, adding to the festive atmosphere.
The war, fought over the exact location of the boundary between Ethiopia and Eritrea, began in May 1998 and left tens of thousands of people dead.
It ended in 2000 with the signing of the Algiers agreement, but peace was never fully restored as Ethiopia refused to implement a ruling by a border commission established by the agreement.
After more than two decades, families divided by the conflict will be able to go and visit each other.
Eritrea won independence from Ethiopia in 1991 but members of the same family continued to live on both sides of the border as the two countries enjoyed good relations until 1998.
The crossing at Zalambessa is on the main trade route linking the capital of Ethiopia's northern Tigray region with Eritrea’s capital, Asmara.
Its closure damaged trade, and consequently the economy in the border region suffered.
The reopening of the border at Burre should allow Ethiopia to access Eritrea’s southern port of Assab.
This is just the latest in a series of rapid changes as relations between Ethiopia and Eritrea have thawed.
In July, Ethiopia’s Prime Minister Abiy and Eritrea’s President Isaias signed a declaration saying that the state of war between the two countries was over.
Since then, phone calls and flights between the two countries have resumed, and last week a ship registered in Ethiopia arrived in an Eritrean port.
The two countries have also reopened their embassies in each other’s capital cities.