"Those run outs were special," Southee said. "They were both great pieces of fielding, always handy.
"We speak about fielding being an attitude, everyone tries hard, and it was nice to see those turn the game.
"The last couple of sessions we created pressure and the last half hour swung it back our way."
The run outs made up for Ross Taylor dropping a catch at first slip from the second ball of the second over. Southee delivered just short of a good length, luring Kusal Mendis into a regulation edge.
Neil Wagner also generated a difficult chance for Taylor off Chandimal who slashed at a ball outside off stump. Taylor parried it over the top. However, the opportunity was probably best catalogued under 'saving four', given Chandimal swung at it like he was wielding a samurai sword.
Taylor also had a couple of chances fall short.
On an emerald wicket, Sri Lankan batsman Milinda Siriwardana said after being sent in their aim was to get to 325 by stumps before rain intervened.
Southee looked the best of the New Zealand bowlers, finding his rhythm early on the way to two for 51 from 16 overs.
With 158 test wickets he is three away from passing Danny Morrison and moving into outright fifth on the all-time New Zealand list behind Sir Richard Hadlee, Daniel Vettori, Chris Martin and Chris Cairns.