England, set 378 to win, finished on 378-3 with Joe Root 142 not out and Jonny Bairstow 114 not out.
Labuschagne, the ninth Australian man to cross the 900-point mark in the ICC's Test batting rankings, is ahead of Root, with Australia's Steve Smith third, New Zealand captain Kane Williamson fourth and India's Rohit Sharma fifth in the latest standings published on Wednesday.
Set 468 runs to win, England was bowled out for 192 with 21 overs remaining on the final day of play in the day-night test at Adelaide Oval.
The South African fast bowler removed opener Mayank Agarwal for 60 and then Cheteshwar Pujara first ball to revive the home team after lunch, and after India made a solid start at SuperSport Park in Centurion.
After heavy defeats in Brisbane and Adelaide, the visitors must win in Melbourne to keep the five-Test series alive, with Australia only needing a draw to retain the urn as holders.
After restricting England's second innings to 297, the Australians needed just 20 runs to clinch the match. They did it in 5.1 overs, for the loss of only makeshift opener Alex Carey (9), with Marcus Harris (9 not out) hitting a boundary to finish off.
Rohit, who has replaced Virat Kohli as the national limited-overs skipper, was earlier named vice-captain of the Test squad scheduled to begin the series on Dec.26 in Centurion. The BCCI said Rohit "sustained a left hamstring injury" during his training session in Mumbai on Sunday.
It is an even harder task at the Adelaide Oval, where Australia's 315 for six to defeat England in 1902 remains the best fourth-innings run chase. England's problems stem from being bowled out, after a batting collapse, for 236 on Saturday in reply to Australia's 473 for nine declared.
Resuming on 94 for four in a chase of 305 to win, South Africa were bowled out for 191, losing their last three wickets in the first two overs after lunch.