McIlroy took the lead at the fourth hole when Bradley made a mess of his approach shot, landing near the greenside bunker.
His pitch out proved to be a tricky one and he left himself with a difficult putt for par. McIlroy had a short putt for a birdie and sank it with aplomb.
The Northern Irishman could have gone further ahead at the fifth, if it was not for Bradley holing a 15-foot putt for a half.
McIlroy did extend his lead at the sixth hole, however, by chipping in from the fringe to the roar of the European fans.
Bradley was able to snatch a hole back with a fine birdie at the eighth.
Both men gave themselves a decent chance of a birdie at the ninth, with McIlroy closer to the cup.