India's women posted a modest total of 96 for seven in their 20 overs at the Feroz Shah Kotla stadium in the capital.
Pakistan had reached 77 for six in 16 overs before their reply was brought to a halt by a heavy downpour which forced the umpires to abandon the game.
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Pakistan were then named the winners under the Duckworth-Lewis method, to the delight of Pakistani captain Sana Mir.
"We have won games by one run, and lost games by a couple of runs so as a unit, we take a lot of pride in our fielding and bowling and this is the way forward," said Mir.
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"It's really good for us to get these pressure games early in the tournament. Very important win for us."
The women's victory came just ahead of the high-voltage clash between the Indian and Pakistani men's teams in Kolkata, which is expected to draw hundreds of millions of television viewers.
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