A sell-out crowd of 28,000 at the Sher-e-Bangla National Stadium braved oppressive heat and humidity to cheer the hosts in their first game since the World Cup in the Caribbean.
The match was reduced to 47 overs a side after light drizzle delayed the start by 50 minutes.
Bangladesh, who defeated India and South Africa on way to their best ever finish of seventh place in the showpiece event, did not disappoint their fans after choosing to take first strike.
Omar and Tamim Iqbal put on 78 for the first wicket before India hit back with two wickets in the space of four deliveries.
Tamim, who smashed 45 off 53 balls, holed out in the deep in left-arm spinner Dinesh Mongia's first over.
Captain Habibur Bashar fell in the next over for zero, mistiming a ball from burly off-spinner Ramesh Powar to give Virender Sehwag a tame catch at mid-on.
Omar and Saqibul attacked the Indian bowlers on the slow wicket to put on 107 for the third wicket.
Omar hit seven boundaries and two sixes - one each off Powar and Mongia - before he was run out by a direct throw from Zaheer Khan while attempting a second run in the 39th over.
Saqibul departed two overs later for a stroke-filled 50 when he was stumped off Mongia to make Bangladesh 200-4.
Bangladesh lost Mohammad Ashraful, Aftab Ahmed and Mohammad Rafique in the last two overs, but still managed to set India a challenging target of 5.3 runs an over.
The Indians, hoping to avenge the World Cup defeat in the three-match series, wilted in the 38 degrees Celsius (100 Fahrenheit) heat with numerous misfields and erratic bowling.
Mongia finished with 3-49 and Powar took 2-55, but seamer Zaheer Khan was hammered for 54 runs in his seven overs.
Bangladesh 250 for 7 in 47 overs (Javed Omar 80, Saqibul Hasan 50, Tamim Iqbal 45, Mohd Ashraful 29; Dinesh Mongia 3 for 49, Powar 2 for 55) Vs India (yet to bat).