The Hoops' 2-1 home win over Spartak Moscow on Wednesday night sealed their place in the last 16 of the Champions League and they continued their good work in Ayrshire, starting when skipper Scott Brown fired them into the lead in the 27th minute.
Second-half goals from midfielder Joe Ledley and substitute Georgios Samaras were just reward for the Parkhead side's dominance, before Cillian Sheridan grabbed a late consolation.
The win keeps Celtic three points clear of second-placed Inverness, who were 3-0 winners at home to Hibernian.
Ross Draper's free header sent Caley Thistle on their way after 13 minutes before Richie Foran and Billy McKay secured their deserved win in the final 14 minutes.
Motherwell recorded their first home victory since the beginning of September as they defeated Ross County 3-2.
Rocco Quinn gave the visitors the lead after 17 minutes, however their advantage was short-lived as Michael Higdon headed home eight minutes later.
Nicky Law then fired the hosts ahead in the opening minute of the second half, but Quinn hammered home an equaliser.
Both sides showed a determination to grab a winner and Keith Lasley clinched the game for Motherwell when he struck with eight minutes remaining.
Hearts beat Aberdeen 2-0 with goals from Ryan Stevenson and Callum Paterson, while nine-man St Johnstone clung on for a 1-1 draw with St Mirren.
The Perth side had Chris Millar and Steven Anderson sent off but, despite the numerical advantage, St Mirren had only Paul Dummett's goal to show for their efforts as the visitors held out for a share of the points thanks to Murray Davidson's early goal.