Sri Lanka, set a target of 120, knocked up the remaining 43 runs in just eight overs after rain washed out over two sessions of play on Saturday.
Resuming on 77 for no loss, openers Upul Tharanga (71) and Michael Vandort (40) batted fluently as Sri Lanka won the series 2-0 after winning the first Test in Chittagong by eight wickets.
Tharanga, who had scored 165 in the first innings, continued his fine form to smash eight fours and one six in his second knock. Other left-hander Vandort was more sedate, hitting four fours off 82 balls.
Sri Lanka had earlier on Friday dismissed Bangladesh for 201 in just over two sessions after gaining an 82-run first innings lead. Six overs remained on the third day when bad light stopped play.
Bangladesh have now lost all their seven Tests to Sri Lanka and their overall record stands at 37 losses from 42 matches with four draws and a lone victory over Zimbabwe.
Bangladesh, who lost a preceding one-day series to Sri Lanka 2-1, take on Kenya in a four-match one-day series at home from March 17.
Result: Sri Lanka win by 10 wickets
Series result: Sri Lanka win two-Test series 2-0
Man of the match: Upul Tharanga (SRI)
Umpires: Asad Rauf (PAK) and Krishna Hariharan (IND), TV Umpire: Mahbubur Rahman (BAN)
Match Referee: Clive Lloyd (WIS)