Goals from Kris Commons and James Forrest helped Neil Lennon's men to victory, with Aberdeen keeper Jamie Langfield seeing red for the home side.
Stuart Armstrong gave the visitors a half-time lead before Gavin Gunning and Kevin Thomson were both sent off. Scott Robertson scored with a long-range strike 10 minutes from time to claim a point for the home side.
St Mirren were held to a 1-1 draw by 10-man Kilmarnock. Kris Boyd gave the visitors the lead before being sent off and the home side soon made their advantage pay when Gary Harkins fired home an equaliser.
Results in full: Aberdeen 0-2 Celtic, Hibernian 1-1 Dundee United, Inverness Caledonian Thistle 2-0 Motherwell, St Johnstone 4-0 Ross County, St Mirren 1-1 Kilmarnock