New Zealand beat Sri Lanka by innings and 58 runs, sweep series 2-0
Having been asked to follow on, Sri Lanka needed 416 runs to make the hosts bat again but were bowled out for 358 in their second innings at the Basin Reserve.
"Tomorrow's weather wasn't looking flash," Southee told reporters of a rainy day five forecast.
"The guys were a bit heavy in the legs but we're very pleased to have it wrapped up with a day to spare."
The inaugural World Test Championship holders, New Zealand's title defence ended long before but they finished the cycle on a high with three straight wins, including a pair of absolute thrillers against England in Wellington and the series-opener in Christchurch.
It was no coincidence that talisman Kane Williamson was key in all three wins.
New Zealand's finest batter was named Man of the Series, having scored 215 in Wellington after steering the Black Caps home with an unbeaten century in the final-ball win in Christchurch.
Henry Nicholls was Man of the Match in Wellington for his unbeaten 200 as New Zealand piled on 580 for four in their declared first innings after Sri Lanka captain Dimuth Karunaratne won the toss and elected to field.
Sri Lanka's hopes of a first win in the country since 2006 all but ended when their batters managed only 164 in reply.
Yet they showed pride and grit to frustrate New Zealand on a day four blighted by severe wind gusts.
A stiff northerly was so strong that the host broadcaster stood a camera crew down for safety, leaving footage from only one end of the pitch.
"The way we bowled I think we are not deserving to win because we bowled so many loose balls and gave the freebies to the Kiwis," Karunaratne told reporters.
"The way the guys batted (today) I'm really proud of that but we have many things to improve."
Though victory appeared well beyond Sri Lanka when they resumed on 113 for two in the morning, there was some hope they might bat out the day and even save the match with the help of day five rain.
After Karunaratne's 51 late on day three, Kusal Mendis (50) and Dinesh Chandimal (62) made fifties, and number six Dhananjaya de Silva ploughed on to the cusp of a century by tea.
However, a wicket before and after the tea-break, including De Silva's dismissal for 98 with a botched scoop off Michael Bracewell, put the game back in New Zealand's hands as Sri Lanka fell to 318 for seven.
Led by a 110-ball vigil for 20 by Kasun Rajitha, the tail did well to thwart New Zealand's bowlers for hours longer than they would have hoped.
Southee (3-51) blew all three of New Zealand's reviews trying to close out the game but the skipper made up for it by taking the last two wickets, with Williamson taking the final catch in the slips to dismiss Rajitha as light waned.
After their calamitous first innings, Sri Lanka batted bravely but five of their top seven batters fell trying to attack short balls, including the skipper, Mendis and Chandimal.
All-rounder De Silva may be especially rueful with his final shot after falling two runs short of his 10th century.
"Those things happen," said Karunaratne.
"Unfortunately he couldn't get the hundred but he batted really well."