In a dramatic run feast befitting the resumption of cricketing ties between the two South Asian rivals, Pakistan made 344 for eight in reply to India's 349 for seven before a capacity crowd of 33,000 at the National Stadium in Karachi on Saturday.
India's left-arm seamer Ashish Nehra, given the task of sending down the final over with nine needed to win, bowled spot-on to concede just three runs and give his team a 1-0 lead in the five-match series.
India, on their first full Test tour of Pakistan since 1989, will also play three Test matches after the Limited Overs series ends on March 24.
Rahul Dravid made 99 and Virender Sehwag hit a quickfire 79 off 57 balls as India, given first strike in perfect batting conditions, recorded their highest One-day total against Pakistan.
But Pakistan responded magnificently and almost set a world record, falling just short of the most successful second innings total in One-day history.
No team had scored more to win than Australia's 330 for seven against South Africa two years ago.
Inzamam led the way with a superb 122 off 104 balls, Yousuf Youhanna made 73, Younis Khan 44 and Abdul Razzaq 27 off just 16 balls.
Inzamam put on 135 for the third wicket with Youhana and 109 off 88 deliveries for the fourth with Younis to lift Pakistan to 278 for three in 42 overs.
Spinner Murali Kartik hit back for India with the wickets of Inzamam and Younis, and Mohammad Kaif took a stunning catch in the deep, but Pakistan kept the spirited run chase on till Nehra stopped them in the final over.
Inzamam was named Man of the Match, but cricket was clearly the winner as both teams headed for the second match at Rawalpindi on Tuesday.
Millions of television viewers on both sides of the border could hardly have expected the drama to follow after India ran up a seemingly impregnable target.
Dravid, who was bowled by the fiery Shoaib Akhtar one run short of his ninth Limited Overs century, helped India surpass their previous best against Pakistan of 316 for seven at Dhaka in 1998.
Sehwag greeted Inzamam's decision to field first by giving his team a flying start.
The free-stroking Delhi batsman, who smashed 14 boundaries and a six, put on 69 for the first wicket with Sachin Tendulkar and 73 for the second with skipper Sourav Ganguly.
Dravid and Kaif (46) built on Sehwag's blitzkreig by adding 118 for the fifth wicket to lift India past the 300-mark.
The match began dramatically when Akhtar conceded 10 runs in the opening over, including two wides and as many no-balls.
Akhtar was not the only player affected by nerves. Off the last ball of the over, Shoaib Malik spilled a simple catch in the covers off Sehwag.
In his second over, Akhtar forced Tendulkar to give a catch at square-leg -- only to see Australian umpire Simon Taufel declare a no-ball.
Tendulkar rubbed salt in the bowler's wounds by top-edging the last ball of that over to fine-leg for six, but fell to Akhtar soon after for 28 when he edged a catch to gully.
Pakistan added to their woes by sending down 20 no-balls and 10 wides, giving India five extra overs to bat.