Opener Jake Weatherald anchored the Redbacks' innings, scoring a fine 103 off 114 balls.
He was overshadowed only by Player of the Match Alex Ross who scored a blistering 110 in 85 balls to set up a formidable total of 6-346 for the Blues to chase.
Despite the return of Mitchell Starc from a foot injury and captain Peter Nevill electing to bowl first on a cloudy morning at the compact Hurstville Oval, they could not stop the threat of the free-scoring Redbacks.
They raced to a 100-run partnership through Carey (60 in 50 balls) and Weatherald before the former was bowled by a fine delivery from Sean Abbott which angled into the off stump.
Cue Ross who was able to score through all parts of the ground, smashing Nathan Lyon for an Australian List A record of 32 runs in a single over.
The Blues looked comfortable in their chase from the outset, with a Daniel Hughes century and a 189-run partnership with Kurtis Patterson.
However, they then lost two wickets in two balls which threw the match back in the Redbacks' favour which included a Mennie hat-trick chance - his second of the tournament.
The menacing quick struck through the latter overs and finished with figures of 4-53 to secure his side victory and first place on the JLT One-Day Cup ladder.
The Redbacks next face the Tasmanian Tigers at North Sydney Oval on Tuesday.