Ashleigh Gardner's quickfire 31 also gave urgency to Australia's innings on Thursday as they made 4-172 after captain Lanning opted to bat first at Newlands.
Scratchy for the most part, save for Gardner's 18-ball blitz, Australia upped the tempo in the last five overs, scoring 59 for the loss of two wickets.
Both Mooney and Lanning benefited from sloppy catching from India, whose dropped chances in the 2020 T20 World Cup final had played a part in Australia clinching the trophy at the MCG.
Lanning, dropped on one and nine, struck six boundaries in her 34-ball knock to bring up her highest score of the tournament.
Her brisk 36-run second-wicket stand with Mooney took the title favourites to 88 in the 12th over, when an offcutter from Shikha Pandey had the left-hand opener cut a dolly to Shafali Verma, who had dropped her at long-on when she was on 32.
Earlier, Alyssa Healy kicked off the high-octane clash with a four first ball in the six-run opening over from swing bowler Renuka Singh Thakur.
The veteran wicketkeeper-batter, who sat out Australia's last group-stage fixture with soreness in her left quad, was on 20 off 21 when India opted for the DRS to overturn a not-out lbw decision but replays showed a pronounced under-edge.
Australia's powerplay score received a boost through a 12-run sixth bowler as Mooney carted the first six of the match off offspinner Deepti Sharma.
Three balls into her opening over, second-change Radha Yadav, replacing Deepti, however, broke the 52 opening stand with Healy's wicket.
The 22-year-old, having come in for fellow left-arm spinner Rajeshwari Gayakwad, dragged her length short to have a waltzing Healy stumped for a scratchy 26-ball 25.
From 0-43 after six overs, Australia hobbled to 1-69 at the halfway mark when they could have well been three down.
Sneh Rana, added to India's main squad of on the morning of the match as a replacement for Pooja Vastrakar, was on her haunches as wicketkeeper Richa Ghosh dropped Lanning when the Australia captain was on one off three balls.
Gardner's arrival injected momentum into Australia's outing with the bat. The No. 4 hammered five fours in her innings.
The 25-year-old fetched 10 runs, including two fours, in the 11-run 17th over upon Thakur's introduction for a second spell but she was bowled by a Sharma yorker next over for 31.
Lanning opened the 20th over from Thakur with a 66-metre six over the deep-backward square-leg boundary off a full toss almost waist-high.
A four tickled fine followed from Lanning before she sent the last ball of the 18-run over flying over square leg for a six, off another high full toss.
With left-arm spinner Jess Jonassen back into the line-up, Australia had put themselves in a strong position to defend successfully.
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