Trent Boult celebrates the dismissal of Sri Lanka's Angelo Mathews during the day two of the second Test match in Colombo on Friday. AP
The pacemen claimed two wickets each as Sri Lanka reached 144 for six at lunch when rain came down and no further play was possible.
Skipper Dimuth Karunaratne moved from his overnight 49 to score 65 and provided the only bit of resistance in the morning session, which saw 29.3 overs bowled.
Boult took two early wickets in one over after Sri Lanka resumed the day at 85 for two. Dhananjaya de Silva on 32 and Dilruwan Perera on five were at the crease.
Southee, like Boult, also delivered a double-wicket maiden as the duo put on a superb display of seam bowling under overcast conditions.
Boult got overnight batsman Angelo Mathews caught behind for two to reach a landmark of 250 Test wickets -- only the third Kiwi bowler to achieve the feat.
Boult trapped Kusal Perera lbw for nought after just three balls as Sri Lanka slipped to 93-4 and would have been in further trouble had Boult held on to a simple catch from de Silva off his own bowling a few overs later.
"Today's a special day for Trent. He's become the third highest wicket-taker for New Zealand," New Zealand bowling coach Shane Jurgensen told reporters.
"He's got great passion and energy which he brings to our team. I think the skill and execution and the way he's contributed to the team has been amazing."
Karunaratne also got a lifeline on 61 when Kane Williamson could not grab a tough chance at short mid-wicket off Ajaz Patel.
But the left-hander, who completed his 23rd Test fifty after resuming the day on 49, soon fell, caught behind off Southee for 65.
Southee struck again in the same over to send Niroshan Dickwella trudging back to the pavilion for nought.
"Tim and Trent have been doing it for a while. They've been creating wickets and they're very accurate, and they've got the ability to swing the ball," said Jurgensen.
"Today there might have been a little bit of swing with the overhead conditions, and it just seemed to be doing a little bit. They both work really hard on their fitness, and that's a big factor in their resilience.
"And they don't just play Test cricket. They play basically all formats. To play all formats and do as well as they've done is awesome."
Perera then played out the rest of the session with de Silva.
Medium-pace bowler Colin de Grandhomme and spinner William Somerville took a wicket each on the opening day after Sri Lanka elected to bat first.
Weather permitting, extended playing time is planned for the third day after two days of continuous rain interruptions in the Sri Lankan capital.
New Zealand skipper Kane Williamson won the toss and elected to bat in the opening Test against Sri Lanka in Galle on Wednesday. The Kiwis return to Cricket action in the World Test Championship
Galle: A fighting half-century by wicketkeeper-batsman BJ Watling helped New Zealand to take a highly competitive lead of 177 at stumps on day three in the first Test match against Sri Lanka in Galle on Friday. New Zealand, who trailed Sri Lanka by 18 runs in the first innings, finished on 195 for seven and a target around 200 could prove tricky
No team have successfully chased more than 99 runs at Galle, giving the visitors hope they can be victorious at the ground for the first time ever after three previous losses.
"I want to invite them to Turin or Lisbon to come to have dinner with me because I want to give something back."
Singapore’s skies were clearer on Friday and air quality improved as smog from Indonesian forest fires drifted away, easing fears that this weekend’s Formula One race may be affected.
Al Jazira kicked off the 2019/20 Arabian Gulf League (AGL) season with a comfortable 2-0 victory over Al Dhafra at Mohamed Bin Zayed Stadium on Thursday. Al Jazira started the game