Sarfaraz-11 had to dismiss Bangladesh for seven runs or lower to qualify for the last semi-final spot—a next to impossible task.
But Bangladesh managed to go past the seven-run mark in the second over, eliminating Pakistan's chances of going through to the next round.
Australia, India, England and New Zealand have now qualified for the semi-finals from the 10-team group phase of the tournament.
The official Twitter account of the ICC Cricket World Cup confirmed that New Zealand had qualified for the semi-finals, knocking Pakistan out of the race for the top four and sending them home.
Earlier, Pakistan had retained a mathematical chance of joining Australia, India and England and denying New Zealand a spot. But that chance was lost when Bangladesh went past the seven-run mark in chase of Pakistan's 316-run target.
A win would still put Pakistan level with New Zealand on 11 points, but their net run-rate would be markedly inferior.
The most realistic way for Pakistan to progress out of a range of unlikely scenarios would have been to score 400 and win by 316 runs. The biggest-ever men's ODI win was by New Zealand against Ireland in 2008, when the margin was 290 runs.
Pakistan were shot out for a paltry 105 in their first match, with the West Indies winning in just 13.4 overs, meaning they took a huge hit on run rate.
They won just one of their first five matches, coming back strongly to beat South Africa, New Zealand and Afghanistan to sit fifth in the 10-team table, but in the end have left themselves too much to do.
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