Southee did a lot of the damage, picking up the wicket of Dinesh Chandimal with his first ball and finishing with the excellent figures of 3-18 from 5.3 overs.
Trent Boult (1-23) and Adam Milne (1-17) also chimed in with wickets on the bowler-friendly wicket to leave Sri Lanka in trouble and Andy Ellis picked up 2-22 in his four overs.
Opener Upal Tharanga put up the most resistance, compiling 60 before he was snared by Southee and captain Mahela Jayawardene made 24 before he was undone by a sharp piece of work in the field by Nathan McCullum.
Sri Lanka won the five-game series 3-0 but neither side has been able to find any rhythm because of the weather and they will both feel underdone heading into the two-test series starting in Galle on Saturday.
"The series so far has been very disappointing and, as a group, we have been very disappointed in the way we have performed,'' Southee said. "We wanted to salvage something out of the [one-day] series and to leave 3-1 would have been nice. It's a very disappointing way for it to end.''
It has gone well for one New Zealander, with BJ Watling named man of the series for his 166 runs at 83.