'To win in Australia is really special, as it really feels like home here,' said Johnson, who has spent a handful of summers in Australia. 'It means so much to me. When I hit the last putt, it changed my life - it was my week.'
The final 18 holes at The Lakes Golf Club featured a tight battle with Johnson, Laycock, Parry and overnight leader Andre Stolz deadlocked with 41 points at the turn.
Although Johnson was the first to stumble when he bogeyed the 10th hole after hitting into water, he was able to rebound with birdies at Nos. 11 and 14. Laycock, meanwhile, birdied the 11th and 15th holes to move one ahead of Johnson at 45 points.
Johnson then picked up a crucial two points by sinking a 20-foot birdie putt from off the green at the 16th. Laycock, playing in the group ahead, hit a poor approach to the par-5 17th to set up a bogey that cost him a point.
Laycock would have run away with the title had he holed makeable birdie putts at the 13th, 14th and 16th holes.
'I missed some real good opportunities on the back nine,' said Laycock, who collected 10 points on Sunday. 'You just can't afford to let those chances go that at this level.'
Parry was also in contention until the end. He notched his fourth birdie of the round at the 17th to climb within one point of Johnson, only to end any hope of a final birdie when he missed the green at the par-3 18th.
Parry then gave up sole possession of second place by missing a putt from inside three feet for his only bogey of the day. He finished alongside Laycock with 44 points after closing with seven points.
'If I talk about it, I'll swear, it is so frustrating,' said Parry, who captured last month's New Zealand Open. 'I played pretty well, but that's just the way it goes.'
Parry came into the ANZ Championship first on the Australasian Tour Order of Merit (money list) but only 86th in the World Golf Ranking. He was counting on a victory this week to lift him into the top 64, which would have earned him a bid in the WGC-Match Play Championship later this month.
Stolz, the leader after the second and third rounds, finished solo fourth with 43 points. He produced just four points in the final round, as did Stephen Gallacher, who was alone in fifth place with 39 points.
Ian Garbutt enjoyed the biggest surge Sunday with 19 points off seven birdies and an eagle, moving him from 45th to sixth place with a total of 38 points.
Jonathan Lomas had trouble building on his opening avalanche of 20 points that gave him the first-round lead and left him 17 points ahead of Johnson, who managed just three points on Thursday. Lomas went on to post one, nine and seven points over the next three days and wound up in seventh place with 37 points.
The modified Stableford scoring system, which was making its first appearance in a European Tour event since 1991, awards five points for an eagle, two for a birdie and none for a par. One point is deducted for a bogey and three are taken away for a double-bogey or worse.
The ANZ Championship was co-sanctioned by the European and Australasian Tours.
Full-field scores from the ANZ Championship