Shakib Al Hasan and Nurul Hasan were fined and given a demerit point apiece for behaviour which the BCB said must not be repeated.
Match referee Chris Broad said the behaviour of Shakib Al Hasan and Nurul Hasan in Colombo was "unacceptable" and they "went overboard".
Bangladesh head into their upcoming T20 tri-series still without injured all-rounder Shakib Al Hasan.
After missing the two Tests between Bangladesh and Sri Lanka, the fit-again Shakib Al Hasan has been named in the Tigers' Twenty20 squad.
Captain Shakib Al Hasan remains sidelined with a finger injury and will miss Bangladesh's second and final Test with Sri Lanka.
With Shakib Al Hasan injured, Bangladesh have added Sunzamul Islam and Tanbir Hayder to their Test squad to face Sri Lanka.
Sri Lanka endured a miserable 2017 on the ODI front, but took tri-series glory against a Bangladesh side who were shorn of Shakib Al Hasan.
Sri Lanka are Bangladesh are set to contest the tri-nations final after the hosts eased to victory against Zimbabwe on Tuesday.
Six years after being succeeded by Mushfiqur Rahim, Shakib Al Hasan is back as Bangladesh skipper.
Shakib Al Hasan's break from Test cricket may hurt Bangladesh in South Africa, but Mushfiqur Rahim hopes there are long-term benefits.