After being out in to bat at the toss, Salt struck 85 from 56 balls while Brook smashed 78 off just 35 deliveries as the visitors amassed a total of 236-4 in their 20 overs.
Th Black Caps could only muster 171 all out in reply with spinner Rashid the destroyer-in-chief with figures of 4-32 in four overs.
Salt and Brook shared in a brutal partnership of 129 from just 69 deliveries while Tom Baton finished off the innings with a quick-fire 29 not out from 12 balls to set a record total for a T20I at the Hagley Oval.
The home team lost wickets at regular intervals in the run-chase as England seamers Luke Wood and Brydon Carse striking early in the innings.
Rashid came into his own in the middle overs as the home side's batters were forced to him the spinner in pursuit of a rising required-rate.
The victory means England go 1-0 in the three-match series with one game left to play after the first match was abandoned due to rain.
Australia took control of the second Ashes Test after the third day following a lower-order batting fightback and then taking six England wickets before the close of play.
Australia went to stumps on the second day of the pink ball Ashes Test on 378-6, the home side leading England by 44 runs on the first innings with four wickets still remaining.
Will Jacks is set to make his first Test appearance in almost three years after England named their starting XI for the second Ashes encounter against Australia that starts in Brisbane on Thursday.
England captain Ben Stokes has come out and said he will accept criticism for his team's performance in the first Ashes Test but has pushed back at jibes that the tourists are "arrogant".
