Pakistan were reeling at 101 for nine in response to New Zealand's 211 before tailenders Mohammad Aamer and Saeed Ajmal launched the unlikeliest of pursuits.
They got within sight, reaching 204 and needing just eight runs off the final over bowled by Jacob Oram. However, No 11 Saeed Ajmal, on 33, skied the first ball to fine leg where Kyle Mills sprinted forward to hold a difficult chance.
Mills was surrounded by relieved and physically exhausted teammates, with the New Zealanders having very nearly lost a match which they seemingly had locked away.
The Pakistan hero was left-handed No 10 Mohammad Aamer, who was unbeaten on 73 off 81 balls.
He launched the rearguard with three sixes in four balls from New Zealand captain Daniel Vettori early in the remarkable 103-run partnership which was the second-highest final-wicket stand in ODI history. The highest was an unbeaten 106 between West Indies greats Vivian Richards and Michael Holding, against England in Manchester in 1984.
The Pakistan pair ensured the run rate never climbed beyond six runs per over, batting with aggression and certainty lacking from nearly every other batsman in a low-scoring game.
New Zealand nerves frayed near the end as the last pair survived a series of miscued shots and appeals for caught behind. Seamer Mills was convinced Aamer had snicked one to wicketkeeper Brendon McCullum in the penultimate over and shared an exchange with the batsman when wandering back to his mark.
Vettori, who had won the toss, said he didn't appreciate the dramatic conclusion.
"Not really, I think we would have liked to have finished it a lot earlier but you have to appreciate the innings that Ajmal and Aamer put on. It showed a lot of determination and really put the pressure back on us.
"I suppose in the end, to defend 212 was an amazing effort."
Earlier, McCullum shone again with 76 off 78 balls at the top to follow his matchwinning 131 in the second match on Saturday.
McCullum's only real support came via fellow-opener Aaron Redmond's 21 and 44 from Ross Taylor.
New Zealand were cruising at 138 for two in the 27th over when McCullum fell, caught and bowled by Shoaib Malik, sparking a slump. The last eight wickets fell for 73 as Pakistan's spinners cast a spell, the visitors dismissed in the 47th over.
Offspinner Ajmal was the pick of the attack, taking four for 33 as the brakes were applied late.
Pakistan's opening stand was 47 but they then lost nine wickets for 54, with many falling to injudicious shots not needed when pursuing a small target.
Oram and Tim Southee were comfortably the most successful New Zealand bowlers. Oram took three for 20 off 9.1 overs while Southee claimed two for 26 off his full complement.
Vettori was pleased his team held their nerve sufficiently in the field to squeeze out a win.
"We wanted our bowlers to lead from the front. Once we put them under pressure with three quick wickets, everyone started to believe.
"Right up until that 10th wicket fell, we were believing most of the time," he said.
"We managed to put some great fielding performances in and in the end that was probably the big difference and the fact that our top order got away.
"Brendon McCullum has been outstanding in this series."
The close nature of the game was a far cry from the first two matches of the series, also at the Sheikh Zayed Stadium.
Pakistan won the first by 138 runs while New Zealand responded with a 64-run victory.
The teams now travel to Dubai for Twenty20 matches on Friday and Saturday (NZT).
