Kohli stamped his authority early in the 202-run chase alongside local favourite Devdutt Padikkal as the duo kept the scoring rate firmly under control, with Bengaluru posting 76-1 in the powerplay and wrapping up the chase in 15.4 overs.
Put in to bat, Hyderabad slipped to 29-3 inside the powerplay as IPL debutant Jacob Duffy took three early wickets, before a resilient partnership between stand-in captain Ishan Kishan and Heinrich Klaasen, backed by lower-order cameos, lifted the visitors to 201-9.
Bengaluru made light work of the target as Kohli and Padikkal punished Hyderabad's bowlers and exposed the void left by the absence of injured ace pacer Pat Cummins.
Kohli, who now features only in one-day internationals, was given a reprieve on 28 before making Hyderabad pay, scoring an unbeaten 69 off 38 balls, including five fours and five sixes as the home crowd roared the team on and chanted his name.
Padikkal was equally fluent, smashing seven fours and four sixes in his 61 off 26 balls, while captain Rajat Patidar added 31 to keep the runs flowing.
Earlier, pacer Duffy dismantled Hyderabad's top order, removing attacking openers Abhishek Sharma (7) and Travis Head (11) in quick succession before also dismissing Nitish Kumar Reddy for one to finish his spell with figures of 3-22.
Kishan top-scored for Hyderabad with 80 off 38 balls, including eight fours and five sixes, while Klaasen added 31 and Aniket Verma blasted a rapid 43 off 18 to give the innings late momentum.
Bengaluru will look to build on their winning start when they host Chennai Super Kings at their home ground on April 5th, while Hyderabad will play Kolkata Knight Riders on Thursday.