Phil DeFreitas earlier scored a career-high 88 as England raced to 328 in its second innings and then skittled out Australia for just 156.
Replacement paceman Lewis took 4-24 and Devon Malcolm 4-39.
Australia still leads 2-1 in the five-match series with the fifth Test to begin in Perth on Friday.
The win was only England's second in the last 23 Ashes Tests.
Australia looked like surrendering meekily at 83-8 until Ian Healy and Damien Fleming frustrated England, sharing 69 runs for the ninth wicket.
The two stayed at the crease for an hour and 52 minutes, frustrating the attack until Lewis had Fleming trapped leg before wicket after making his highest Test score of 24.
Healy remained unbeaten on 51 when last man Peter McIntrye was dismissed by Malcolm for a duck.
Malcolm and Lewis accounted for five of the seven wickets to fall during the post-lunch session after DeFreitas had provided the backbone of the victory with his aggressive 95-ball innings.
Malcolm lived up to his true potential as a tearaway fast bowler by firing out skipper Mark Taylor (13), Michael Slater (4) and Steve Waugh (0).
Taylor was caught at first slip -- Slater hooking a bouncer to Phil Tufnell at fine leg -- and two balls later Steve Waugh's off stump was uprooted as Australia crashed to 23 for four in the space of 22 deliveries.
This was after David Boon (4) had pulled a bottom edge off an Angus Fraser delivery to wicketkeeper Steve Rhodes.
Mark Waugh and first innings centurian Greg Blewett figured in a brief stand -- 41 runs in just under an hour.
Mark Waugh fell to left arm spinner Phil Tufnell's first delivery -- Mike Gatting catching him off a rebound from his own left instep.
After finding the edge of the groping bat of Blewett (12) -- Lewis trapped Shane Warne's last ball before tea and completed the over forcing Craig McDermott to edge a low catch to Rhodes.
DeFreitas earlier came out of the shadows of his overnight partner John Crawley (71) -- caning the Australian pacemen in front of a sparse 12,074 crowd.
He was particularly severe on fast bowler McDermott -- hooking, pulling and driving four fours off five deliveries -- the first of them to reach 50.
DeFreitas completed the over, the bowler's 27th, with a towering six over mid-wicket as he took 22 runs from the six balls.
In just 82 minutes of batting, England raised 128 runs, but lost the last three wickets for just 11 runs as Mark Waugh mopped up the innings with his best test haul -- five for 40.
Crawley completed an elegant half-century and extended the seventh-wicket stand to 88 runs before miscuing a pull to sky a return catch to Waugh.
Crawley batted for 212 minutes for his 152-ball innings, which included five boundaries.
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