Norman's big finish, that included a 30-foot putt on the 18th, snatched victory from Hart as he closed with a final-round five-under 66 and claimed his 17th career PGA Tour victory and first of the year.
The Australian finished 16 strokes below par with a 268 total at the rain-affected event, overtaking Hart, who closed with consecutive 66s and was in the club-house for an agonising 90 minutes with a two-stroke lead at 15-under as Norman mounted his late assault.
Norman, who had a miserable Masters and US Open this year, will enter next month's British Open with his confidence bolstered by the victory. "Winning is a great tonic," Norman said.
Norman collected $270,000 (pounds 168,00) for the victory, surpassing $11m (pounds 6.8m) in career earnings.
Craig Parry and the rookie Robert Damron, who led after three rounds, finished tied for third at 14-under 270.
Seve Ballesteros, out of action with back trouble since quitting the Volvo PGA championship six weeks ago, begins his comeback at this week's Murphy's Irish Open.
Despite his health worries and his dismal 1997 record - nine missed cuts in 11 starts which has seen him drop to outside Europe's top 160 this season - Ballesteros still has not given up hope of being a playing captain when Europe defend the Ryder Cup in Spain in September.
"I will try to make the team and depending on how I feel and what the situation is, I will make the decision that is in the best interests of the side," he said in an interview in this week's first issue of Golf International magazine.
Ireland will host a regional qualifying competition for the Open championship for the first time next year. County Louth has been chosen as one of the 14 venues. Other new choices are Blackmoor in Hampshire and Stockport, while Wildernesse in Kent is brought back into the list.
Those dropping off after staging qualifying events next Monday are North Hants, Northants County, Sundridge Park and Wilmslow.
1998 OPEN CHAMPIONSHIP Qualifying list: Beau Desert, Blackmoor, Burnham and Berrow, Carlisle, Copt Heath, County Louth, Coxmoor, Glenbervie, Hankley Common, Moortown, Romford, South Herts, Stockport, Wildnernesse.
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