Dettori came through on the outside with 300 meters to go and came home unchallenged from 6-1 shot Eagle Mountain, the mount of Johnny Murtagh and one of eight runners for Irish trainer Aidan O'Brien.
The Heat's cause was not helped by the poor free throwing of Shaquille O'Neal, who missed eight straight free throws before finally hitting one in the final moments of the game.
The South African carded eight birdies and one bogey to overtake Miguel Angel Jimenez of Spain and Australia's Nick O'Hern at the top of the leaderboard.
The choice proved a sensible one as the Munster home pack sent number eight Anthony Foley over for a try before O'Gara's conversion went over via an upright.