Needing to restrict Sri Lanka to a score of 147 or less in a 213-run chase, the home team reached 207 in their 20 overs as captain Dasun Shanaka almost pulled off the victory.
The Sri Lanka skipper smashed an unbeaten 76 from just 31 balls, including three sixes in the final over, but the hosts fell agonisingly short of the target.
The result means New Zealand join England and South Africa in the last four of the T20 World Cup while India and West Indies face off on Sunday for the final spot.
After being sent in to bat at the toss, Pakistan got off to a brilliant start as opening pair Sahibzada Farhan and Fakhar Zaman put on 176 inside 16 overs.
Fakhar struck 84 off 52 balls while Sahibzada reached exactly a 100 from just 60 deliveries.
Sri Lanka fought back to take further seven wickets in last four overs to restrict the visitors to 212-8 in their 20 overs.
Sri Lanka had looked dead and buried in the run-chase when they slumped to 101-5, even leaving New Zealand nervous watching on, but Pavan Rathnayake with 58 kept his team in he game.
Shanaka's brilliance then saw the home outfit almost pull off the heist but seamer Shaheen Shah Afridi held his nerve to see out the contest.
The Proteas will look to keep up their "discipline and intensity" in the final T20 World Cup Super 8s clash against Zimbabwe in Delhi on Sunday, batting coach Ashwell Prince has said.
Will Jacks says England are relishing life under Harry Brook's captaincy after booking their place in the semi-finals of the 2026 T20 World Cup with a perfect Super Eights record.
England completed an unbeaten Super 8s phase at the T20 World Cup after a thrilling four-wicket victory in the final over against New Zealand in Colombo on Friday.
Tilak Varma insists India want to continue to play 'fearless cricket' in their must-win clash against West Indies at the T20 World Cup despite home pressure.
