The neighbours said they would take measures to avoid a repeat of "untoward" incidents in May, which saw gunfire exchanged between both countries' security forces, the Myanmar Police and Border Guard Bangladesh said in a joint statement.
The unrest comes amid Myanmar assertions that an insurgent group claiming links to the Rohingya Muslim minority had started to operate in the area.
Waves of sectarian conflict two years ago in Rakhine State left the region largely segregated on religious grounds.
Myanmar provided information about the activities of the Rohingya Solidarity Organisation (RSO) "inside the territory of Bangladesh which are detrimental to peace and stability along the border areas", according to the statement.
Bangladesh also affirmed it "always show zero tolerance on the issue of miscreants/illegal armed groups".
The US State Department says the RSO was active on the border in the early 1990s after a Myanmar military crackdown that caused tens of thousands of Rohingya to flee into Bangladesh.
But experts say it has long been viewed as a defunct armed force.
"They are not a new group," a senior police official told AFP, adding that the RSO were viewed as "extremists", but declining to give further information about their supposed activities.
Myanmar police officials have told AFP that they believed security forces had been attacked by members of RSO in the days leading up to the death of the Bangladesh soldier on May 28.
Gunfire erupted again on the border two days later during negotiations for the return of the man's body.
Myanmar has rejected Bangladesh claims that the dead soldier was a legitimate member of its border security force and Thursday's joint statement made no mention of the fatality.
Both sides agreed to greater cooperation between border security forces to combat illegal armed groups, criminals and insurgent groups operating along the border. They also agreed to deploy border liaison offices to better manage border affairs and reduce cross-border criminal activity.