The visitors started well as they took the game to Celtic and created two great chances with Shaun Maloney having to clear a John Sutton header off the line and goalkeeper Mark Brown was also forced to save from David Clarkson.
Celtic responded at the other end as Scott McDonald broke down the left and played a pass in to Scott Brown, whose shot crashed away off the post.
Motherwell dominated possession for long spells without taking the lead they probably deserved despite Sutton and Chris Porter both coming close.
Celtic made a tactical switch with Barry Robson moving wide to allow Scott Brown to move inside.
This seemed to lift the home side and Cillian Sheridan fired an angled drive at goal which Graeme Smith did well to turn away for a corner.
Just as the game was heading to half-time, Motherwell only half cleared a Sheridan cross and it fell kindly for Paul Hartley, who took a touch before firing a low shot past Smith.
The early exchanges of the second half saw both sides create chances with Stephen Hughes and Porter going close for Well and Sheridan heading wide for Celtic.
The game then turned in the 58th minute when referee Dougie McDonald decided that Clarkson's flailing arm which caught sub Paul Caddis was enough for a straight red card.
Ten-man Motherwell's challenge was buried 13 minutes later when McDonald latched onto a loose ball at the edge of the box to rifle a shot low past Smith.
Celtic were then allowed to relax on their two-goal cushion and saw out the remainder of the game with relative ease.