Three ahead overnight, the 41 year old Ulsterman stretched that to four with a 15 foot birdie putt on the opening green.
But Molinari, the elder of the two Italian brothers who are both contending for the first prize of more than €600,000 and are Ryder Cup contenders this year, came back at him in superb fashion.
After a 40 foot birdie putt at the second the 29 year old struck a fairway wood to six feet on the next and eagled.
Clarke birdied the hole, but when he bogeyed two holes later - only his second dropped shot of the week - his advantage was down to one.
Both birdied the long sixth and they were level when Molinari made a 30 footer at the seventh and Clarke missed from six.
The tee on the ninth had been moved forward to make the par four driveable, but Molinari, needing a birdie to turn in a blistering 30, could not match his playing partner's birdie and so Clarke, out in 33, started for home one in front at 13 under par.
Molinari's brother and Omega Mission Hills World Cup winning partner Francesco, who has a golden opportunity to move from ninth to fifth in The Ryder Cup standings this weekend, was up into third place on eight under when he birdied the short eighth and sank a 20 foot putt for an eagle two at the next.
One stroke further back was Swede Peter Hedblom and in fifth spot was Shane Lowry, the 23 year old who won last year's 3 Irish Open as an amateur.
Clarke's compatriot Graeme McDowell, in his first European Tour event since winning the US Open Championship, re-ignited hopes of a third successive victory when he covered his first 13 holes in five under to stand five under.
But then came bogeys on the 14th and 16th.
Clarke looked in danger of being brought back level on the short 11th after missing the green, but he saved par from 12 feet while Molinari three-putted, failing to hole from four.
However, on the 560 yard 13th Clarke ran up a bogey six after his chip failed to make it up the bank in front of the green.
He was 12 under, but Molinari missed a seven foot birdie chance and was one behind.
Hedblom had eagled the hole and went third on nine under.