Neil Wagner grabbed two wickets as the hosts set up the possibility of a result inside three days after initially facing the prospect of a tough target to chase on days four and five.
Southee grabbed his fourth wicket when Nuwan Pradeep was caught behind for a three-ball duck to end the innings.
The visitors had been comfortable until about 30 minutes before lunch having cruised to 71 without loss.
Doug Bracewell then grabbed two wickets in an over, dismissing Dimuth Karunaratne (27) and Udara Jayasundera (0) with short deliveries.
Jayasundera's dismissal was a controversial one as umpire Paul Reiffel initially ruled not out but New Zealand called for a review and the 'hot spot' technology revealed a dull mark on his gloves as the ball passed.
Third umpire Richard Kettleborough asked Reiffel to overturn his decision, much to the bemusement of both Reiffel and Jayasundera. None of the other technology had indicated the ball had hit the batsmen's gloves.
Wagner grabbed another wicket in the penultimate over before lunch when he had the dangerous Dinesh Chandimal caught for four at leg gully by Martin Guptill.
The hosts then did more damage after the break, taking seven wickets in 8.1 overs, with only some counter-attacking blows from Milinda Siriwardana, who blasted three fours and two sixes in his 18-ball innings of 26, padding Sri Lanka's score.
All 10 wickets fell for 62 runs in a little over 80 minutes.
The hosts had resumed on Sunday on 232-9 before Trent Boult was caught by Rangana Herath off Nuwan Pradeep at mid-on for a duck. Dushmantha Chameera finished with career-best figures of 5-47 from 13 overs.
(Reporting by Greg Stutchbury in Wellington; Editing by Peter Rutherford)
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