“Moulin Rouge!,” based on the 2001 pop-mashup movie musical, bested “Jagged Little Pill” and “Tina: the Tina Turner Musical” for the evening’s most coveted statuette. The production collected 10 awards, including for direction, choreography, actor, supporting actor, set design, costumes, lighting, orchestrations and sound. A new play version from London of the chestnut “A Christmas Carol” was runner-up, with five awards.