South Africa came into the contest with high hopes of a first World Cup crown after an excellent run in the tournament but had no answer to the power hitting of New Zealand at the top of the order.
The Black Caps won the toss and elected to bowl, restricting the Proteas to what looked a below par 169 for eight in their 20 overs.
They made light work of their reply, reaching their target for the loss of only Tim Seifert (58 from 33 balls) as they attacked from the opening over in a comprehensive display of batting.
Allen was superb and smashed eight sixes to take the game away from a South African side with no answer to his destruction.
"I wanted to get into good positions and put in a good performance for the team," Allen said. "We wanted to start looking straight and put them not he back foot early.
"It was a huge game for us, and Tim and I keep each other in it when we are on strike or not. You adapt to the wicket, training is important to get used to the conditions.
"You take the positives from this game and celebrate the success, and then look forward to the final on Sunday."
The Proteas had played most of the tournament in Ahmedabad, where the quicker wicket suited their batters. Kolkata has a bit more grip and it was a struggle from the start.
They lost Quinton de Kock (10) and Ryan Rickelton (0) in consecutive balls, while skipper Aiden Markram was subdued with 18 from 20 deliveries.
Dewalt Brevis (34 from 27 balls) tried to launch a counterattack, but it was only an unbeaten 55 from 30 balls by Marco Jansen that got them to anything like a respectable score.
The New Zealand spinners Mitchell Santner (0-25 in four) and Rachin Ravindra (2-29 in four) kept the scoring rate down, while fast bowler Lockie Ferguson (1-29 in four) was also impressive.
"I think you look at the conditions and they bowled really well up front," Markram said. "Credit to their bowling unit. And obviously when someone plays an innings like that, you don't often come out on the right side of it.
"To get to 170 was a great effort to be fair. As it goes in T20 cricket, they got off to a flyer in the powerplay. Massive credit to Finn Allen's knock and Seifert's knock to kill the game off as early as they did.
I think we expected the wicket to play really well, looked really good to the eye. Maybe we had to try and scrape our way to 190 and we'd be in the game. Obviously disappointed in the result but really proud of these guys.
"We played so well in this tournament. I think we'll let the emotions settle first and foremost and then have a discussion as a group. You get back on the horse and come back. Hugely disappointed. We'll have to come out stronger and be better as a team."
New Zealand, who are coached by South African Rob Walter, will meet either India or England in Sunday’s final, hoping to go one better than their 2021 defeat to Austyralia in the decider.
