The Black Caps will take on South Africa on Thursday at Eden Gardens in the semifinal clash.
Meanwhile India beat the West Indies by five wickets, with four balls to spare, to set up a match against England in the other semifinal.
The 2024 runners-up, South Africa, chased down a target of 154 with 13 balls to spare to remain the only unbeaten side in this year's 20-team tournament.
Having already qualified for the semifinals, South Africa rested speedsters Kagiso Rabada and Marco Jansen and spinner Keshav Maharaj and still restricted Zimbabwe to 153-7 at the Arun Jaitley Stadium.
Zimbabwe skipper Sikandar Raza smashed 73 and returned to claim 3-29 but even his all-round brilliance was not enough to halt the South African juggernaut.
"Nice to get the win," South Africa captain Aiden Markram said.
"It can be quite tricky when you change the team a bit.
"Obviously with the bat, I'd have loved it to be more clinical, but probably not the worst thing that the middle order was put under a little bit of pressure and came through."
South Africa began with George Linde's left-arm spin following Raza's decision to bat after winning the toss.
Zimbabwe's top order struggled but Raza counter-attacked in spectacular fashion and raced to a 29-ball 50.
Raza clobbered four sixes and eight fours in his 43-ball blitz before falling to Kwena Maphaka.
Clive Madande made 26 not out down the order but Zimbabwe still settled for a below-par total.
Raza did not throw in the towel, however, and led Zimbabwe's fightback with the ball.
Armed with the new ball, the off-spinner dismissed Quinton de Kock and rival captain Markram in successive overs to jolt South Africa out of any sense of complacency.
Ryan Rickelton smacked four sixes in his 31 but Zimbabwe struck just when a partnership was about to bloom.
Likewise, Dewald Brevis (42) threatened to run away with the game but Raza returned to dismiss the batter to keep Zimbabwe alive in the contest.
Tristan Stubbs (21) and Linde (30) combined in an unbroken stand of 53 to guide South Africa to victory.
"I've never got a man-of-the-match (award) after losing a game but I think I can hold my head high," Raza said.
"We took a bit of beating in the previous two games but with each beating came a valuable lesson.
"We have improved with every game that we have played. At least we now have got a taste of what it needs to be done for us to be a mid-table team."
New Zealand reached the Twenty20 World Cup semifinals after Pakistan crashed out despite a five-run victory against already-eliminated Sri Lanka in a Group Two Super Eights match in Pallekele yesterday.
An inferior net run rate meant Pakistan had to win big against Sri Lanka in order to pip New Zealand and join group leaders England in the last four.
They racked up 212-8 after Sahibzada Farhan (100) and Fakhar Zaman (84) combined in a T20 World Cup record 176-run opening stand.
- Reuters/RNZ
