Tipped as one of the favourties this time around, South African middle-order crumbled while chasing a modest 222 win as they were all out for 172 in 43.2 overs at the Sher-e-Bangla stadium.
The coveted trophy thus continues to elude South Africa, which has now failed to win a match in the knockout stage of the World Cup for the fifth time in a row.
South Africa produced a disciplined bowling effort to restrict New Zealand to a modest 221 for eight after being asked to bowl but the total proved too much for them.
Jesse Ryder (83) and Ross Taylor (43) saved New Zealand blushes with a crucial 114-run stand for the third wicket and later Kane Williamson (38) played a good hand towards the end to help the Kiwis put something on board for a fight.
New Zealand then showed a lot of heart in defending the small total with both spinners and pacers bowling exceptionally well under pressure.
Nathan McCullum, who started the proceedings for New Zealand with his off-spinners and struck in the very first over, starred with three wickets, including that of key batsmen Hashim Amla and JP Duminy. He gave away just 24 runs.
Oram (4/39) though was pick of the bowlers while Tim Southee (1/44) took the crucial wicket of Jacques Kallis (47).
Faf du Plessis (36) gave the Kiwis a scare with his gritty knock towards the end but it was New Zealand`s day out there in the end.
New Zealand will take on the winner of tomorrow`s match between England and Sri Lanka, in the semifinal on Tuesday in Colombo.
Kallis at one stage was steering his side comfortably towards the target but once he was dismissed, South African batting collapsed dramatically.
They lost the last eight wickets for just 64 runs and the slide started when Kallis was caught by Jacob Oram near the deep mid-wicket boundary off Southee.
South Africa`s chase also started in a bizarre dismissal of in-form Hashim Amla.
A confident looking Amla got out in a freak fashion as an under-edge off Nathan thudded on the foot of stumper Bredon and flew towards first slip Daniel Vettori, who accepted the catch with glee.
Kallis joined his skipper Smith at the wicket and he did the bulk of the scoring initially as he hit Vettori and first change bowler Tim Southee for three boundaries.
Smith was not fluent with his stroke-making but was ticking the board with ones and twos. He hit Jacob Oram for a cover boundary to raise his side`s 50 in exactly 10 overs.
With no success coming, Vettori introduced spinner Luke Woodcock but Smith welcomed him with another cover boundary.
As Smith was gradually getting into his elements, Oram cut short his stay at the crease by having him caught at point with substitute Jamie How being the fielder. Smith raised 61 runs with Kallis for the second wicket.
Kallis and Ab de Villiers (35) kept South Africa in hunt with their 39 run partnership but Kallis dismissal changed the complexion of the game.
Nathan bowled Duminy and de Villers was run out in a matter of the three deliveries.
Oram dismissed Johan Botha (2) and Robin Peterson (0) in successive overs and New Zealand did not let South Africa make a comeback after that.
Earlier, Morne Morkel (3/46) was the pick of the South African bowlers while Dale Steyn (2/42) and Pakistan-born leg-spinner Imran Tahir (2/32) snared two wickets apiece to stop the Kiwis way behind the 250-run mark.
For Kiwis, Ryder starred with the bat with a gritty 83 off 121 balls during which he struck just eight boundaries. He added a vital 114 runs with Taylor for the third wicket to help the Kiwis recover after they were reduced to 16 for two at one stage.
Towards the end, Williamson came up with crucial 41-ball 38-run knock to take New Zealand beyond the 200-run mark.
It was not an ideal start for New Zealand after winning the toss as they lost both their openers -- Brendon McCullum and Martin Guptill cheaply inside the first six overs.