Captain Ricky Ponting and Andrew Symonds hit 58 runs each to help the world champions recover from a disastrous 4-2 to post 240-9 after being sent to bat in the day-night semi-final.
New Zealand were bowled out for 206 in reply after 36-year-old Glenn McGrath dispelled doubts over his future in the game with a superb 3-22 in an unbroken opening spell of 10 overs.
The Black Caps were in danger of falling short of their lowest one-day total of 64 against Pakistan in Sharjah in 1986 when they slipped to 35-6 by the 15th over.
But Daniel Vettori (79) and Jacob Oram (43) gave the Aussies a fright by adding 103 for the seventh wicket as the late evening dew made it difficult for bowlers to grip the ball.
Symonds broke the threatening stand by having Oram stumped in the 36th over before and Vettori was bowled by Mitchell Johnson in the 43rd.
New Zealand added 171 runs for the last four wickets amid full-throated support from some 20,000 Indian fans, but the effort was not enough to secure victory.
It was Australia's 16th win over their trans-Tasman rivals in the last 18 meetings, sending them to Sunday's final in Mumbai against the winner of Thursday's other semi-final between South Africa and the West Indies in Jaipur.
Australia had not made the final in the four previous editions of the biennial tournament despite winning the last two World Cups in 1999 and 2003.
"It's pleasing to finally get there," said Ponting. "I thought we batted well and it was a total which we knew we could defend.
"I am not surprised New Zealand fought back because they are a tough side, but we were confident that we would pull it off.
"It does not matter who we play in the final. Both South Africa and the West Indies have a good record against us in recent times so we will be ready."
New Zealand captain Stephen Fleming said it was "poor" batting by the top order that cost his side the match.
"The Australians bowled well with the new ball but we batted very poorly at the start," he said. "We were outskilled at the top and must develop batsmen who can face the heat with the new ball."
Ponting's men were stunned by hostile Kyle Mills before they pulled themselves out of a deep hole at the Punjab Cricket Association ground.
Mills, the 27-year-old from Auckland, finished with 4-38, but the New Zealand batsmen undid his good work in the second session.
McGrath had Lou Vincent caught at second slip by Ponting for one and Brett Lee bowled Nathan Astle for zero as the Black Caps slipped to 20-2 by the seventh over.
Two more wickets fell in eight balls at the score of 30 to diminish New Zealand's chances of causing an upset.
McGrath forced Hamish Marshall to edge a catch to wicket-keeper Adam Gilchrist, before Fleming fell in Nathan Bracken's first over.
Fleming, his team's main hope in the absence of the injured Scott Styris, made 15 off 28 balls when he hung his bat out to Bracken and was snapped up by Ponting in the slips.
McGrath then bowled Peter Fulton for two and had Brendon McCullum caught by Damien Martyn to leave New Zealand struggling at 35-6.