They also bowled and fielded well to take the six-time champions into the final over in a record run chase.
Beth Mooney provided a cool head for Australia as it came to the final over needing eight runs to win. She struck a four from the first ball from Jhulan Goswami, who was playing in her 200th one-day international, and another boundary from the third ball to clinch the win and finish 30 not out.
Lanning had been lacking runs in earlier matches but was player of the match for the way in which she measured Australia's run chase.
"Personally it was nice to contribute," Lanning said. "I was disappointed with the last couple of games but really felt at training I flicked the switch a little bit about how I wanted to play. It was nice to be able to execute and contribute to a win. It was a belter of a wicket. It came on really nicely, it didn't turn and any width given was going to be hard to defend."
India's captain Mithali Raj also was her team's top-scorer with 68 and Yastika Bhatia and Harmanpreet Kaur made centuries as India attempted to set a target which was beyond Australia's reach.
But Healy and Rachael Haynes (43) put on 122 for the first wicket to lay the foundation for Australia's run chase and a 103-run partnership for the third wicket between Elysse Perry (28) and Lanning took them closer to victory.
After a rain delay, India pegged back their opponents and dismissed Lanning in the penultimate over.
But with eight runs needed from the final over, and Australia curiously not using star finisher Ash Gardner, Mooney was calm under pressure, completing the highest chase in World Cup history.
- With AP