The 38-year-old veteran put in an impressive all-round performance, taking 4-31 before making 24 to help the home side take an unbeatable 2-0 lead in the three-match series.
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 19 overs to spare. Jayasuriya was named man of the match.
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 caught behind, a decision which left the wicketkeeper-batsman clearly unhappy.
Mahela Jayawardene (seven) was caught by Ahmed at midwicket after the Sri Lankan skipper mistimed a pull off Syed Rasel, the most impressive Bangladesh bowler, with figures of 2-39 from 10 overs.
Tillakaratne Dilshan (20 not out) and Chamara Silva (36) put on 57 runs for the fifth wicket to ensure their team's emphatic win.
Earlier, Sri Lanka rode on fine bowling performances by Lasith Malinga and Jayasuriya after they won the toss and elected to field.
While Malinga (3-22) inflicted the early damage, Jayasuriya (4-31) polished off the tail with his slow left-arm bowling.
Malinga, the bleach-blond paceman with a low-slung round-arm action, bowled a fiery opening spell to send back left-handed openers Shariar Nafees (six) and Tamim Iqbal (three) before returning to dismiss Mushfiqur Rahim for a duck.
Ahmed provided the only meaningful resistance, top scoring with 47 off 85 balls, with two fours.
Jayasuriya hastened the slide when he removed Bangladesh skipper Mohammad Ashraful (29), who put on 61 runs for the fifth wicket with Ahmed.
The third and last one-dayer will be played here on Wednesday.