Ashleigh Gardner's 21-ball 29 also helped Australia overcame a slow start in the Cape Town showdown on Sunday.
Mooney's quickfire 15-ball stand of 33 with Ellyse Perry (seven off 5 balls) ensured the defending champions scored 46 in the death overs despite Shabnim Ismail's double-wicket 20th over.
The 34-year-old South Africa quick became the leading wicket-taker in women's T20 World Cup history with her 42nd strike.
She was on a hat-trick when Tahlia McGrath took a single off the final ball as Australia finished on 6-156, leaving the match evenly poised.
Earlier, Gardner, promoted to captain Meg Lanning's usual No.3 slot, marshalled Australia from 1-36 in the powerplay to double that run tally by the halfway mark.
Her 41-ball, 46-run second-wicket stand with Mooney, who played out a maiden in the sixth over against Ismail, broke upon the introduction of part-time spinner Chloe Tryon.
The left-armer had Gardner hole out to long-off, where South African captain Sune Luus completed a high catch without a hitch.
The allrounder then took a superb catch at deep midwicket off a Lanning pull to reduce the opposition to 4-122 with 17 balls left in the innings.
Australia amassed 34 off those 17, with Mooney, who brought up her second successive fifty of the World Cup, off 44 balls, alone fetching 25 in that tally.
Mooney's pivotal innings rescued Australia after her fellow opener, Alyssa Healy, scrapped to a 20-ball 18 that ended with a catch at cover off Marizanne Kapp.
Australia's chances of a successful defence of the total in the final, and their title, will rely as much on their pacers as on the spin attack, led by Jess Jonassen and Georgia Wareham.
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