In took the Bhoys only two minutes to break the deadlock. Lee Naylor swept in a left-wing cross which was nodded across the penalty area by Jan Vennegoor of Hesselink for Gravesen to crack the ball past Tony Bullock.
Two free-kicks from Shaun Maloney failed to add to the scoring but Gravesen scored his second midway through the first half when he ambled through the St Mirren midfield, before dispatching an 18-yard shot which was deflected on its way past Bullock.
Saints replied doggedly but had no penetration in attack and another Maloney free-kick forced Bullock to make a low save. Kenny Miller fell under a Kirk Broadfoot tackle but the referee ignored strident penalty claims. Saints forced the game's first corner a minute before half-time but Celtic cleared comfortably.
After bizarre advice from the stadium announcer about sectarian singing - which was not issued when Rangers visited recently - Artur Boruc blundered to allow John Sutton in but the Polish keeper blocked both Sutton's and Billy Mehmet's shots.
Hugh Murray was unlucky to be booked for winning the ball against Vennegoor of Hesselink.
Simon Lappin exploded into magnificent form on the left flank and, from his free-kick, Mehmet headed against a post.
Celtic failed to contain Lappin and, within a minute, the flanker's superb cross was headed in spectacularly by the stooping Sutton, leaving the much-vaunted pairing of Stephen McManus and Gary Caldwell looking inept.
Celtic's first and only corner - awarded erroneously by the assistant referee - resulted in Gravesen's header from Shunsuke Nakamura's kick crossing the line despite Garry Brady's attempt to clear.
That knocked the stuffing out of Saints and Maciej Zurawski was inches away from number four with a low angular drive.