Southee had Kamrul Islam trapped in front for two in the 85th over, shortly after the game had been delayed for about 10 minutes when Rubel Hossain received medical treatment after being hit on the elbow by Trent Boult.
Southee finished with figures of 5-94, while Boult backed up his new ball partner with 4-87.
Souyma Sarkar, promoted to open in a patchwork Bangladesh side who were missing batsmen Imrul Kayes and Mominul Haque and captain Mushfiqur Rahim due to injury, top scored with 86 after the visitors had been asked to bat by Kane Williamson.
Soumya and first Test double centurion Shakib Al Hasan began to counter attack.
The pair were not afraid to play their shots and raced through to 128 for two with numerous boundaries of the 15 the duo struck before lunch flying just wide of the hands of fielders down to the third man or fine leg fence.
Boult finally broke the 127-run partnership after lunch when Soumya was caught by Colin de Grandhomme in the slips.
Nurul Hasan, deputising for Mushfiqur, and debutant Nazmul Hossain steadied the innings and took their side through to 232 before Nazmul was caught by Jeet Raval off Southee for a patient 18.
Mehedi Hasan Miraz (10), Taskin Ahmed (eight) stayed with Nurul for as long as they could as he crept towards his first Test half century, but he was the ninth wicket to fall for 47 when Watling took his fourth catch of the innings.
New Zealand won the first Test by seven wickets.