Left-hander Cook struck a superb 102 on only his sixth appearance, silencing his critics who questioned the suitability of his game to the limited-overs arena
Bell, playing his 48th game, reached the elusive three figures, compiling an unbeaten 126 off just 118 balls. It was an innings of the highest quality on a day when England showed a welcome return to form in the shorter format.
Cook and Bell, whose second-wicket partnership of 178 allowed Kevin Pietersen the freedom to hit 33 not out off 25 deliveries late in the innings.
England lost Matt Prior early after being sent to bat. Prior was caught at mid-on as he attempted to work Zaheer Khan through the leg side, moments after hitting the same bowler back over his head for six, but India had to wait more than 30 overs for their next success.
Cook mixed his usual fondness for the on side with a clutch of sweetly-struck boundaries through cover off the back foot, and continued to rotate the strike sensibly despite seeing opening partner Matt Prior depart for 19.
Cook's confidence and range of stroke grew as he and Bell gradually upped the run-rate, the former swatting young leg-spinner Piyush Chawla to wide long-on for four shortly after passing his previous best, the 41 he made against Sri Lanka at Headingley Carnegie last year.
A 74-ball fifty arrived shortly after, and he was followed to the landmark by Bell, whose straight six off Chawla remained the highlight of his innings.
Neither Cook nor Bell offered a chance as they partnership passed the 150 mark, and the opener's delight was obvious when he went to a 122-ball century containing eight fours.
He eventually fell to RP Singh, playing on as he advanced down the wicket and swinging to leg, but Pietersen maintained the momentum without ever truly freeing his arms.
He helped add 67 in little more than eight overs, accompanying Bell, whose previous highest score was 88 against Pakistan last year, to the three-figure mark with a leg glance for four off Ajit Agarkar.
Only Zaheer and RP bowled with the purpose and their opening spells were economical and testing. However Indians bowled waywardly in the middle and end overs concedeing too many easy runs.
England 288 for 2 in 50 overs (Bell 126 n o, Cook 102, Pietersen 33; RP Singh 1 for 49, Zaheer 1 for 49) Vs India