Sciver’s century stand with Lydia Greenway, who made 53, helped England recover from 80-4, before both departed with the victory line in sight.
Greenway hooked a full toss straight to midwicket before Sciver picked out mid-off with 22 runs required to win.
Georgia Ellwiss and Katherine Brunt saw off those runs as England took the two points on offer from the first one-day international.
Ellyse Perry had top-scored with 78 in Australia’s 238-9 after the tourists had won the toss and elected to bat.
In response, England’s top order all got starts but failed to kick on. Charlotte Edwards (15) dragged on facing Perry before Heather Knight picked out Meg Lanning at midwicket for 12.
Sarah Taylor looked impressive until the 20th over when, on 30, she nicked behind off Erin Osborne, who had accounted for Amy Jones (15) in her previous set of six.
Australia scented blood but Greenway and Sciver’s combination helped England towards victory.
Australia’s score was thanks largely to Alex Blackwell and Perry. The pair came together at the crease with the score 35-3, after Brunt, who took 3-48, and Kate Cross had accounted for Australia’s top three.
Blackwell and Perry put on a stand of 121 that saw both post fifties – Perry’s her sixth in as many one-day internationals.
The stand was broken by Heather Knight, whose shy at the stumps from mid-off sent Blackwell on her way for 58.
Perry, who was dropped on 21 by Becky Grundy, went on to 78 before picking out Sciver on the deep midwicket boundary. A direct hit from Edwards accounted for Osborne either side of Taylor combining with Greenway and then Brunt for further run-outs as Australia stuttered to a close.
England captain Charlotte Edwards said: “I thought we started brilliantly with the ball. Our fielding was outstanding at times and then that partnership between Sciver and Greenway was absolutely brilliant. And for Georgia Elwiss to come in and show her talent at the end was fantastic.”