David battered a 26-ball fifty after coming to the crease with his team 5-33 in the seventh over of a truncated match. But his teammates mustered 45 off 58 balls as RCB were restricted to 9-95.
Hazlewood then took 3-14, including the wicket of Josh Inglis (14), but was unable to prevent Kings winning with 11 balls to spare. Victory was wrapped up when Marcus Stoinis, recalled for the out-of-touch Glenn Maxwell, smoked his second ball for six over mid-wicket.
Rain, then a wet outfield, delayed the start and reduced the match to 14 overs apiece. Within four overs RCB had lost Phil Salt, Virat Kohli and Liam Livingstone for 26 runs, the latter to Xavier Bartlett (1-26).
Skipper Rajat Patidar made 23 but no one else apart from David reached double figures and RCB had slumped to 9-63 by the time Hazlewood joined him with 13 balls left.
With David farming the strike, and hitting three successive sixes, the pair added 32, Hazlewood facing one ball, a RCB record for the tenth wicket.
Marco Jansen took 2-10, Yuzvendra Chahal 2-11 and Arshdeep Singh 3-23 off their three overs each.
Curiously, RCB did not open with Hazlewood and Kings got off to a solid start being 1-32 before the Australian reduced them to 4-53. But Nehal Wadera cracked 33 off 19 balls to ease any nerves and RCB were never realistically in the game.
Kings have not qualified for the play-offs since 2014, when, captained by George Bailey and with Maxwell in commanding form, they topped the regular season table before losing in the final.
Finishes of eighth and ninth in the last two seasons have led to another overhaul with Ponting brought in as head coach and Shreyas Iyer an expensively-recruited captain.
Besides Maxwell, Stoinis, Inglis and Bartlett they also have Aaron Hardie on the playing staff, Brad Haddin as Ponting's assistant and James Hopes as bowling coach. So far their Australian-accented approach is working.
