The defending champions lost their tournament opener by nine wickets to New Zealand but they showed no signs of a hangover in Dharamsala.
Jos Buttler's men made a steady start after being asked to bat at the picturesque Himalayan ground but upped the pace to reach 100 in the 16th over.
Dawid Malan, who has three one-day international hundreds in 2023, outscored his aggressive opening partner early on, reaching his fifty off 39 balls, with seven fours and two sixes.
But Bairstow found his rhythm, bringing up his own half-century in 54 balls, including eight fours.
It was the first time he has passed fifty in ODIs since returning from a broken leg.
But the 34-year-old was bowled by Bangladesh captain Shakib Al Hasan for 52, to leave England 115-1 in the 18th over.
