Thanks largely to Brook's career-best unbeaten 114 off 49 deliveries, the hosts set Sri Lanka a mighty target of 255 after wrapping up their innings at 4-254 on Tuesday (Wednesday AEST).
The visitors were soon struggling, ending the powerplay 3-32 after Pathum Nissanka, Kamil Mishara and Lahiru Udara all went cheaply.
The rout continued with a steady flow of wickets as Sri Lanka reached 5-71 at the halfway mark under the floodlights, compared with England's 1-134 at the same stage.
Sri Lanka knew they were doomed long before they started the final stand at 9-114, and were 135 all out in 19 overs.
Sonny Baker, playing on his home Southampton ground in his second T20 international, had a standout evening with the ball, finishing his stint with figures of 3-12.
"I thought the performance tonight was near perfect," said Brook, named player of the match after England beat Sri Lanka for the consecutive 13th time in T20 internationals.
England debutant Aneurin Donald, replacing the injured Phil Salt, hit the first six of the evening at the Utilita Bowl off his fourth ball, surviving an lbw call on his second, but was caught for 12 off five balls after Sri Lanka won the toss and opted to bowl on a dry pitch.
Brook replaced him at the crease and the rout began in an electric 128-run partnership with Buttler that left the visitors wondering what had hit them.
England were 1-90 at the end of the powerplay, their second highest in T20s, and Buttler reached his 50 off 29 balls.
Brook scored 30 runs off one over - a six followed by two fours, two sixes and another four off Dunith Wellalage - and chalked up his half-century off a mere 20 balls.
Buttler's breathtaking stint ended at 80 off 44 deliveries, with Wellalage taking a simple catch off legspinner Wanindu Hasaranga as England were 2-159.
The hosts were 3-195 when Tom Banton (10 off 10 deliveries) was caught by Dasun Shanaka off Dushmantha Chameera.
Brook was unstoppable, celebrating his second T20 international century and also equalling the second-fastest ton by an England player in the format.
He ended with his highest score in T20 internationals, 114 from 49 deliveries, featuring 10 fours and six sixes.
"It was a highly impressive all-round performance, but Sri Lanka were very poor," commented former England captain Eoin Morgan on Sky Sports television.
England can seal the three-match series with victory in the second game in Cardiff on Thursday.
