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McIlroy used a 7-iron to connect from 170 yards and ensure he owned bragging rights for the round over playing partner Chris Wood of England.
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Moynihan made three birdies and a bogey in his opening two-under 70 while Decottignies-Lafon kept it clean on the back nine as he picked up four shots after making the turn at two-over.
Former world number 1 says both Australians are in great form
World number one and local favourite Jordan Spieth led from start to finish against Jamie Donaldson of Wales in a 3&2 win.
The former world number one has plummeted to 467th in the rankings having not picked up a club competitively after undergoing two back operations in September and October.
The Royal Golf Mohammedia Open by HLO, starting on Wednesday, will see players from 23 countries step up to the plate and challenge the South African prodigy for supremacy this season.
The Sharjah Senior Golf Masters will form part of the European Senior Tour Order of Merit, with a sizeable prize fund of $450,000 on offer.
The European Senior Tour made its first foray in the UAE in 2000 when the UAE capital hosted the Abu Dhabi European Seniors Tour Championship.
McIlroy made four birdies and no bogeys in the third round on the Blue Monster, his 4-under 68 getting him to 12 under for the week and three shots ahead of Adam Scott and defending champion Dustin Johnson with 18 holes left in the first World Golf Championships event of the season.