Set 106 to win after restricting Pakistan to 105-8 in their 20 overs, India were looking comfortable at 61-1. But two wickets in two balls by Pakistan captain Fatima Sana opened the door before Kaur took her side to the brink of victory.
With India two runs short, however, she slipped and appeared to rick her neck, leaving the field unbeaten on 29 from 24 balls.
Sajeevan Sajana hit the winning runs from the next delivery as India closed on 108-4, winning with seven balls to spare.
"We have been very disciplined, followed the plans," said India's Smriti Mandhana. "In the field we were very good. Better start with the bat would have been good, but we will take that win."
Nida Dar top-scored for Pakistan with 28 from 34 balls but there was little else to cheer after Sana had won the toss and elected to bat on a wicket that produced some awkward bounce.
"We were not up to the mark in the batting. We were 10-15 runs short," said Sana.
All of the Indian bowlers contributed with Arundhati Reddy the most succesful as she removed Omaima Sohail (3) and Aliya Riaz (4) in the middle order before clean bowling Dar in the final over to finish with 3-19 from her four overs.
"I've worked a lot on my T20 bowling in all stages of the innings. I'm still working hard now. I kept it simple and attacked the stumps," said Reddy who was named Player of the Match.
India lost Mandhana for seven but Shafali Verma and Jemimah Rodrigues took them to 61 before the second wicket fell in the 12th over.
Verma, their impressive 20-year-old opener, struck three fours in a 35-ball 32 before holing out to mid-on.
That gave Pakistan a sniff of a chance but Kaur made sure India made it 13 wins from 16 against Pakistan in T20Is.
After losing their opening match to New Zealand, this was a much-needed win for India in Group A. They remain fourth in the group, with Pakistan just ahead of them on net run rate.
India face Sri Lanka in their next game in Dubai on Wednesday while Pakistan meet defending champions Australia on Friday.
West Indies face Scotland in Sunday's late game in Dubai with both sides having lost their Group B openers.
