The pair's opening stand was the highest of the series against South Africa and only the second time New Zealand's openers have passed 50 in the southern hemisphere summer.
Guptill and Brendon McCullum posted 124 against Zimbabwe in Napier in January.
Guptill was on 28, while Flynn, who was struck at least twice on the body as he tried to duck, weave and bob his way past the short-pitched bowling, was on 35 after the first full day's play in the weather-hit test.
The first two days of the match had been badly affected by rain with more than seven hours play lost, while both days also ended early due to bad light.
South Africa captain Graeme Smith had declared the visitors' first innings closed at 474 for nine shortly after tea, with Mark Gillespie taking six for 113.
Vernon Philander (29) was the final wicket to fall when he was caught on the boundary by Flynn to give Gillespie his sixth victim, following up his five for 59 in the second test in Hamilton.
Alviro Petersen top scored with 156, while JP Duminy scored 103 for the visitors, who look to have batted New Zealand out of the test and series.
South Africa have a 1-0 lead in the three-match series following their nine-wicket victory in the second test at Seddon Park in Hamilton, where they bowled New Zealand out for under 200 in both innings.
South Africa had resumed the day on 246 for two with Petersen on 96 and Duminy on 76 with the pair wasting little time to bring up their centuries.
The 31-year-old Petersen, who had accumulated just 66 runs at an average of 16.5 in four test innings in New Zealand, punched a Chris Martin delivery through the covers for two then cut him square for his 13th boundary.
Duminy followed him to the milestone shortly afterwards when he flicked a Gillespie delivery behind square for his second test century.
His celebrations were short lived, however, as he went hard at a good length delivery in Gillespie's next over and was caught by Ross Taylor at first slip, just after he and Petersen had brought up their 200-run partnership.
Duminy was the only wicket to fall in the first session, before New Zealand had some success after lunch, at one stage they took four wickets for 42 runs, before Mark Boucher and Philander steadied the innings.
South Africa upped their tempo after tea with Boucher caught by Kane Williamson in the gully for 46 before Philander holed out on the boundary to force Smith to end the innings.
Scoreboard
New Zealand won the toss and chose to field
South Africa first innings (overnight 246-2)
G Smith c van Wyk b Bracewell 5
A Petersen lbw b Martin 156
H Amla c van Wyk b Gillespie 63
JP Duminy c Taylor b Gillespie 103
AB de Villiers b Martin 38
J Rudolph c van Wyk b Gillespie 11
M Boucher c Williamson b Gillespie 46
D Steyn c Guptill b Gillespie 0
V Philander c Flynn b Gillespie 29
M Morkel not out 10
Extras (w-3, nb-3, b-6, lb-1) 13
Total (for nine wickets dec, 148.4 overs) 474
Fall of wickets: 1-13 2-106 3-306 4-362 5-381 6-388 7-404 8-459
Did not bat: Marchant de Lange
Bowling: Martin 28-5-95-2; Bracewell 30-3-106-1 (w-1); Gillespie 33.4-7-113-6 (w-2, nb-2); Vettori 42-11-98-0; Brownlie 2-0-20-0; Williamson 13-1-35-0
New Zealand first innings
D Flynn not out 35
M Guptill not out 28
Extras (lb-1, nb-1) 2
Total (for no wickets, 25 overs) 65
Still to bat: Brendon McCullum, Ross Taylor, Kane Williamson, Dean Brownlie, Daniel Vettori, Kruger van Wyk, Doug Bracewell, Mark Gillespie, Chris Martin.
Bowling: Steyn 7-3-11-0; Philander 6-0-30-0 (nb-1); Morkel 6-3-6-0; de Lange 4-0-14-0; Duminy 2-0-3-0
South Africa lead the three match series 1-0.
