Latham said winning the toss and putting South Africa in to bat was critical for New Zealand's success.
The tourists crumbled in their first innings for 95 on a green wicket with Matt Henry bagging seven for 23.
A century by Henry Nicholls and a string of 50-plus partnerships saw New Zealand amass 482 in reply and the writing was on the wall for South Africa.
"Fingers crossed we can continue the momentum into the second Test match," Latham said with New Zealand eyeing a first-ever series win against South Africa.
They achieved the victory in their first Test since 2008 without two of their greatest batsmen -- Ross Taylor retired and Kane Williamson injured -- and with strike bowler Trent Boult on paternity leave.
"It's been a long time since we haven't seen any of those three names who were a vital part of this side, but I guess it's testament to the depth we're creating," Latham said.
A frustrated Proteas captain Dean Elgar could not explain the drop off in performance since South Africa won a series against India last month.
"It's something I'm trying to wrap my head around," he said.
"But I do feel our intensity was lacking during the last two and a bit days.
"We were totally outplayed by a classy New Zealand outfit.
"It's extremely frustrating being the captain and the ball is being hit (by New Zealand) on both sides of the wicket. You can't set a field for that."
The tourists lost three wickets for only four runs at the start of their second innings and as the Test extended into a third day, only the diminutive Temba Bavuma and Kyle Verreynne offered any real resistance.
The 1.62-metre (5 ft 4 in) Bavuma, 22 overnight, scored 41 and shared in a 41-run stand with Verreynne before he was trapped leg before wicket by Neil Wagner.
Verreynne fell in the following over for 30, caught in slips off Southee who then quickly snagged the wicket of Kagiso Rabada for zero.
The day started badly for South Africa with Bavuma's overnight partner, Rassie van der Dussen, out on the second ball when Henry got the delivery to nip back, finding a gap between bat and pad.
Zubayr Hamza, who top-scored with 25 when South Africa made a paltry 95 in their first innings, only managed six off 32 deliveries before he edged Kyle Jamieson to Daryl Mitchell at first slip.
The remaining batsmen -- Marco Jansen, Glenton Stuurman and Duanne Olivier -- contributed 21 between them.
Southee finished with five for 35 while man-of-the-match Henry and Wagner took two wickets each.