The wicketkeeper-batsman was 161 not out as the Warriors closed the first day on 5/381 on a typically batter-friendly WACA pitch.
Bancroft shared an unbroken 108-run stand with captain Mitch Marsh, who was unbeaten on 71 from 90 balls.
Kane Richardson was the pick of the bowlers on his return to the side, with 1-59 and four maidens in a tough day for the bowlers.
His scalp arrived in the form of allrounder Marcus Stoinis, when he edged behind to 'keeper Alex Carey.
Shaun Marsh was the first wicket to fall, chopping on from the bowling of Joe Mennie for 17.
WA win the toss and elect to bat, while Richo comes back into the side! #SheffieldShield— West End Redbacks (@WestEndRedbacks) November 13, 2017
Squad ➡️ Weatherald, Dalton, Ferguson, Head (c), Lehmann, Cooper, Carey, Mennie, Richardson, Sayers, Worrall 🕷️ pic.twitter.com/jHiUPCFpxd
Then in the middle session, Cartwright went for 35 when he was bowled by Chadd Sayers.
WA suffered a blow before the game with left-arm quick Jason Behrendorff ruled out with lower grade back soreness. He joins Nathan Coulter-Nile on the sidelines.