It was Simon Lappin who scored the crucial goal for the visitors by coolly steering the ball home in the 33rd minute.
However, a shocking decision from the referee Calum Murray when he sent off Bob Malcolm in the 51st minute left the visitors facing an uphill battle.
The midfielder's tackle on Massimo Donati barely merited a yellow card let alone a red one, though the reaction of the Italian may have influenced the referee.
By the end of the game there were muted chants against manager Gordon Strachan, though most of the home support continued to give him their support.
Celtic could have had this game won in the opening ten minutes with Scott Brown curling an effort narrowly over. Donati cleverly weaved along the byline, but instead of cutting the ball back to the unmarked Scott McDonald, he shot from an acute angle allowing Graeme Smith to save.
Aiden McGeady was giving Paul Quinn a torrid time on the left flank and his cross was headed against the bar by Jan Vennegoor of Hesselink.
Celtic keeper Artur Boruc swung and missed an attempted clearance from Gary Caldwell's pass back, but fortunately for the keeper the ball rolled past the posts.
Motherwell's goal came after a Stephen Craigan took a quick free-kick which reached Ross McCormack. His cross found Mark Fitzpatrick whose shot was palmed out by Boruc, but only as far as Lappin to guide the ball home.
Celtic sought the equaliser before half-time but Shunsuke Nakamura and McDonald missed good chances.
After Malcolm's dismissal it seemed only a matter of time until Celtic equalised, but Smith performed heroics in the Well goal in particular when he touched over a Andreas Hinkel shot.
The home crowd were unimpressed when top scorer McDonald was replaced by Georgios Samaras and it certainly seemed a curious decision by Strachan.
McGeady fired in two powerful shots which Smith saved well, but Celtic's endeavours lacked subtlety and Motherwell claimed a famous victory in their chase for third place.