Australia, sent in to bat on a seamers' wicket, were reduced to 4-2 by Kyle Mills before the world champions pulled themselves out of a deep hole at the Punjab Cricket Association ground.
Mills, the 27-year-old from Auckland, finished with 4-38 as the Kiwis were set a target of 4.8 runs an over to make the final with an upset win.
Mills struck twice in the third over of the innings when he had Shane Watson caught at mid-off with the second ball and Adam Gilchrist was snapped up at square-leg off the fifth.
Damien Martyn (26) helped Ponting add 66 for the third wicket when he fell to a marginal leg-before decision off spinner Daniel Vettori in the 20th over.
Ponting moved to his 55th one-day 50 in the 27th over, but fell two overs later when he holed out to mid-on against Mills.
Michael Hussey, who put on 65 for the fifth wicket with Symonds, went in the 41st over when he slashed left-arm seamer James Franklin to Hamish Marshall at point.
Symonds' run-a-ball knock boosted Australia's lower order before he was bowled by a Shane Bond full toss trying to flick the ball to leg.
Bond, showing no signs of the hamstring niggle that hampered him in the league match against Pakistan, also bowled Brett Lee and ran out Mitchell Johnson with a smart throw on his follow through.
Australia retained the same side that defeated hosts India in the last league match here on Sunday.
New Zealand were without all-rounder Scott Styris due to a back injury.
The second semi-final between South Africa and the West Indies will be played in Jaipur on Thursday.