Molinari stormed into contention with two eagles in the last seven holes to set the clubhouse target of nine under par, which Dunne was able to match with a birdie on the last.
The players returned to the 18th for sudden death and a scrappy par was good enough to give Molinari the win after Dunne found trouble off the tee and failed to get up and down from a greenside bunker.
Dunne’s runner-up finish moved him from 66th to 20th in the R2D standings and effectively – even at this stage of the season – means he has regained his full tour card, surpassing the total amount won in his rookie season last year.
“I felt like there were just so many shots that slipped away from me,” admitted Dunne. “It’s just one of those days. I felt like I was really close to shooting four or five under and stuck with one under in the end. I’m happy with my form and looking forward to next week. I’m just a bit disappointed not to come out with the result we wanted.”
In 10 tournaments played so far this season, Dunne has been in the money in eight and has two top-10s.
Molinari’s previous victories both came in 2010 and the second, in the final qualifying event for the Ryder Cup, persuaded European captain Colin Montgomerie to give the former US Amateur champion a wild card for Celtic Manor.
A serious wrist injury meant Molinari played just 14 events in 2013 and, although he finished 40th on the money list the following year, he had to regain his playing rights via the qualifying school in 2015 and 2016.
“It’s fantastic,” Molinari said. “It’s been a while. I’ve been through some very hard times and to get this I’m very very pleased.
“Last time I made a big putt on 17 and this time I made a big putt on 18 so it looks like I always want to keep it to the last few holes.”
After starting the day four shots off the lead, Molinari covered the front nine in level par before following a birdie on the 11th with an eagle on the par-five 12th after a brilliant approach from 250 yards.
The 36-year-old’s chance looked to have gone when he bogeyed the 16th after finding two bunkers on the tough par four, but he birdied the 17th from close range and holed from 20 feet for another eagle on the last in a superb 68.
Dunne held a two-shot lead overnight and was three clear after recovering from a bogey on the first with birdies on the second and fifth, but bogeyed the sixth and was never able to shake off the chasing pack.
The 24-year-old, who shared the 54-hole lead in the 2015 Open at St Andrews while still an amateur, carded two more birdies on the back nine but immediately followed each with a bogey.
Another dropped shot on the 16th left him needing to birdie one of the final two holes to catch Molinari and a superb chip from the edge of the 18th green set up a tap-in for a closing 72.