The hosts scored at either end of the first half before a stirring Celtic comeback, but it was Motherwell who had the last word, getting their equaliser right at the death.
Motherwell started brightly and were ahead within just four minutes thanks to Sutton, who collected a long pass and slotted the ball past Fraser Forster.
Celtic continued to control the ball and push forward but found themselves two down just before the break thanks to a competent finish from Zaine Francis-Angol.
There was somehow still time for the away side to get themselves back into the match though, as Scott Brown found Anthony Stokes inside the box, allowing the striker to halve the deficit on the stroke of half time.
Celtic didn't take long to get themselves level in the second half, with Stokes again involved. This time the striker set up his partner Giorgos Samaras to fire the Scottish champions level.
Brown and Beram Kayal both tested Gunnar Nielsen as Celtic continued to pour forward, and their pressure eventually told with four minutes to go when substitute Leigh Griffiths found space at the back post and fired them in front.
The hosts weren't done though, as with the final kick of the game, Sutton collected Lionel Ainsworth's pass and earned his side a point.