While Malinga (3-22) inflicted the early damage, Jayasuriya (4-21) polished off the tail with his slow left-arm bowling to claim 300 one-day wickets in 397 matches.
Sri Lanka, who won the toss and elected to field at the Premadasa stadium, lead 1-0 in the three-match series.
Malinga, the bleach-blond paceman with a low-slung round-arm action, bowled a fiery opening spell to send back the top three Bangladeshi batsmen with just 13 runs on the board.
The tourists did not recover from the early blows and lost wickets at regular intervals.
Malinga dismissed the left-handed opening pair of Shahriar Nafees (six) and Tamim Iqbal (three) in the space of two overs before returning to claim the wicket of Mushfiqur Rahim for zero.
Middle-order batsman Aftab Ahmed provided the only meaningful resistance, top scoring with 47 off 85 balls and two fours.
Ahmed added 61 for the fifth wicket with skipper Mohammad Ashraful, who scored 29, the pair successfully milking the spinners after the quick bowlers were removed from the attack.
But Jayasuriya hastened the slide when he removed Bangladesh skipper Mohammad Ashraful (29), who put on 61 runs for the fifth wicket with Ahmed.
Jayasuriya claimed his 300th wicket when he had last batsman Syed Rasel caught by Jehan Mubarak at mid-off.
Bangladesh 137 all out in 46.5 overs (Aftab Ahmed 47, Mohammad Ashraful 29; Malinga 3 for 22, Jayasuriya 4 for 31) Vs Sri Lanka.