The opening pair are widely viewed as two of the best white-ball batters in the world and while they have both had quiet tournaments so far, they flourished when it mattered most against the Black Caps.
A bit of luck was involved as Babar was dropped off his first ball before going on to register 53 off 42 deliveries, taking the sting out of a run-chase of 153 alongside Rizwan in a 105-run first-wicket stand.
Rizwan departed for 57 off 43 balls with 21 still required off the last three overs but there was to be no dramatic twist despite New Zealand's best efforts as Pakistan got home with five balls to spare.
Shan Masood's single off the first ball of the final over made sure Pakistan will head to the MCG on Sunday to face either India or England, who play the other semi-final at Adelaide on Thursday.
Just one win separates Pakistan from the most unlikeliest of tournament triumphs. They lost their first two matches of the campaign and only got out of the group after the Netherlands upset South Africa.
Even losing the toss at a raucous SCG did not hamper them as they restricted New Zealand to 152 for four, with Shaheen Shah Afridi recording excellent figures of 2-24 from his four overs.
The left-arm quick was clattered down the ground for four from his first ball but his second thudded into Finn Allen's pads, with an lbw verdict overturned on review after the detection of a thin edge.
But Afridi's third ball swung back in and clattered the front pad of the dangerous Allen, who reviewed again but this time to no avail as the noise went up a level from Pakistan's vocal fans.
Pakistan were sharp in the field, as evidenced by Shadab Khan's direct hit to run out Devon Conway, while New Zealand struggled to build any sort of momentum in admittedly tricky batting conditions.
Boundaries were proving hard to come by for New Zealand, who registered just 10 fours and a six in their 20 overs, with captain Kane Williamson again chewing up balls without really accelerating.
There was intent in the closing stages but he was deceived by Afridi's off-cutter and bowled round his legs after shuffling across his stumps and missing an attempted scoop to depart for 46 off 42 balls.
Daryl Mitchell registered his second successive 50 in a T20 World Cup semi-final, diligently working his way to 53 off 35 balls, but Babar and Rizwan got off to a flyer in pursuit of the total.
Babar, the former top-ranked batter in the world, made just 39 runs in five innings in the group stage and his lean run almost continued as his first ball grazed the outside edge off Trent Boult.
But Conway, diving to his right, grassed the chance and Babar crashed Boult through point in the left-armer's next over, which proved expensive as Rizwan also took two off-side boundaries.
In total there were nine fours - five for Rizwan and four for Babar - as Pakistan raced to 55 without loss at the end of the powerplay before spin duo Mitchell Santner and Ish Sodhi dried things up.
Pakistan were still in control though as Babar reached his half-century first and Pakistan were into three figures when the captain lofted Boult to long-on as New Zealand had a belated breakthrough.
Rizwan also went to his 50 but, after a couple of tidy overs, the opener lashed a full toss from Boult to the sweeper on the off-side as it seemed some nerves were setting in.
But Mohammad Haris, dropped on 11 and 28 by Santner and Sodhi, thrashed Lockie Ferguson through the covers and over deep square-leg for six.
After Haris was out for 30 off 26 balls, Masood sealed victory with a drive down the ground for one.