Jayasuriya, who has 12,108 runs from 397 one-dayers, took his 300th wicket when he dismissed last man Syed Rasel as Bangladesh were bowled out for 137 in 46.5 overs.
Sri Lanka overhauled the target with almost 19 overs to spare.
Jayasuriya was named man of the match.
"It is nice to be the first one to get there but I am happy I took four wickets and helped my team win," said the left-hander, who has played more one-dayers than any one else in history.
"Such records come when you play for so long."
The World Cup runners-up hardly broke sweat in their chase despite losing Jayasuriya and Kumar Sangakkara (six) in the space of three overs.
Jayasuriya's knock was cut short when he played a short one off left-arm spinner Abdur Razzaq into the hands of Aftab Ahmed at backward point.
Sangakkara was adjudged