India were bowled out for 119 in 19 overs. Bumrah’s 3-14 in four overs proved decisive as Pakistan ended on 113-7.
Despite a tricky batting surface, a record attendance of 34,028 for an international cricket match in the USA witnessed a tight contest.
Naseem Shah's valiant effort in the final over couldn't secure victory, as Pakistan fell short by six runs.
Bumrah dismissed Babar Azam (13) and Mohammad Rizwan (31), and conceded just three runs in the penultimate over.
Pakistan needed 18 from the final over. Arshdeep Singh dismissed Imad Wasim with the first ball, making the chase even tougher.
This defeat leaves Pakistan without a point, with crucial matches against Canada and Ireland ahead to reach the Super Eight stage.
Rain delayed the start of the game, played in front of a packed 34,000 fans, mostly backing tournament favourites India.