The Indians reached their final target of 102 from 22 overs with 17 balls to spare in bright evening sunshine after dismissing Pakistan for 165 from 39.4 of their allotted 40 overs.
Opening batsman Shikar Dhawan, who scored 114 and 102 not out in the first two group matches, again top-scored for the winners with 48 from 41 deliveries.
The final match in Group B had no bearing on next week's semi-finals as India had already qualified while Pakistan were eliminated after losing their first two matches.
There was still plenty of pride at stake, though, in what was only the third competitive match in England between the two sub-continent rivals. All 24,000 tickets were sold within three hours of going on sale in November.
On the day it was the Indian supporters who were celebrating after Mahendra Singh Dhoni had won the toss and elected to field.
Nasir Jamshed (2) was caught at second slip by Suresh Raina pushing forward to a full delivery from paceman Bhuvneshwar Kumar.
Wicketkeeper Kamran Akmal, promoted to open in an attempt to bolster the fragile Pakistan top order, and Mohammad Hafeez took the total to 50 from 12 overs before the players went off for a brief rain break.
On their return Hafeez (27) nibbled at the first ball of the 13th over from Kumar and was caught behind by Dhoni, diving to his right.
Akmal then followed for 21 attempting to drive off-spinner Ravichandran Ashwin. He got an inside edge which flew off Dhoni's left thigh to Virat Kohli at leg-slip.
Ashwin was getting significant turn and Dhoni called up a short-leg to add to the pressure when the rain came again and the players left the field for more than 2-1/2 hours with India 70 for three off 19 overs.
After the resumption Ravindra Jadeja struck a blow by bowling Pakistan captain Misbah-ul-Haq for 22.
Misbah had scored 96 not out and 55 in his previous two innings.
Jadeja, who took seven wickets in India's first two matches of the tournament, finished with two for 30 from eight overs and Ashwin snapped up two for 35 from eight overs. The last two batsmen were both run out without scoring.
India's initial target was 168 from 40 overs based on the Duckworth-Lewis method for rain-affected matches. It was reduced to 157 from 36 following another interruption and finally to 102 from 22.
"The whole team is doing very well in all three departments," Dhoni said.
Kumar was named man-of-the-match after taking two for 19 in eight overs.