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In the other last-eight tie played Saturday, the West Indies beat England by 18 runs in a match shortened to 36 overs a side by bad weather.
India could only manage 114-9 in their 23 overs but Pakistan had to battle to reach the modest total, getting to their target with only three balls remaining and two wickets in hand.
Mandeep Singh top-scored for India with 40, after the 2008 champions lost two wickets in the first over. The main bowlers to cause the damage for Pakistan were Fayyaz Butt with 4-27, and Raza Hassan, who took three wickets.
Pakistan's chase started badly. They lost their first three wickets for just 17 runs in the first seven overs but a partnership of 55 between Ahsan Ali and Rameez Aziz brought them back into the reckoning.
Wickets fell regularly until Hammad Azam steered Pakistan home, scoring an unbeaten 21 from 13 balls.
In the other quarter-final, the West Indies scored 166 before dismissing England for 148.
The West Indies' victory was built on a 103-run third-wicket partnership between Kraigg Braithwaite (69) and Andre Cleary (52), as well as a devastating spell of bowling by paceman Jason Holder, who took 5-19.
Pakistan will meet the West Indies in the first semi-final on Monday. New Zealand will play Australia and South Africa meet Sri Lanka in the remaining quarter-finals on Sunday.
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