Australia hit back to take the lead in the ninth minute before Michaela Blyde looked to secure another final victory for New Zealand with two minutes remaining.
New Zealand were in a strong position to close it out, holding possession deep in Australian territory before losing skipper Hirini to a bloody nose.
Australia then earned a penalty with time up and 90 metres away from the tryline but worked their way up the field to set up the match-winner for Dick.
It was an especially meaningful result for Australia after they were knocked out of the Olympics tournament at the quarter-final stage by Fiji, missing out on defending their title in what would have been a semifinal clash against New Zealand.
"It's awesome to have New Zealand back in the series. We've been looking forward to playing them for such a long time, after obviously not matching up in Tokyo," Australian skipper Charlotte Caslick said after the final.
"That was an awesome effort by the girls. We started well and obviously finished really well."
New Zealand will have their chance for revenge with the World Series concluding in Toulouse later this month and again at the Commonwealth Games in Birmingham in July-August.
Earlier in the day, Portia Woodman scored her 200th World Series try for New Zealand as the Black Ferns Sevens coasted into the semifinals with a 38-0 win over Canada.
They were then tested by France in a rematch of the Tokyo Olympics gold medal match, coming back from 14-7 down at halftime to book a spot in another final appearance after second half tries to Blyde, And Risi Pouri-Lane.