Goals from Sutton and Zaine Francis-Angol had put Stuart McCall's side 2-0 up, with Gunnar Nielsen saving a Kris Commons penalty before Francis-Angol doubled their lead.
Anthony Stokes then pulled a goal back in first-half stoppage time and the champions looked set to pull off a stunning victory when goals from Georgios Samaras and Leigh Griffiths put them 3-2 up with two minutes remaining.
Griffiths, whose future at Celtic has come under scrutiny as he faces Scottish Football Association charges for alleged derogatory chants about Hearts and their former midfielder Rudi Skacel, was unable to end a miserable week on a high though, as Sutton had the final say in time added on.
The draw leaves Motherwell three points behind Aberdeen in the race for second place and seven better off than fourth-placed Dundee United, who were beaten 2-0 at their Scottish Cup final opponents St Johnstone.
Steven Anderson and the prolific Stevie May were on target as Tommy Wright's side secured a victory that leaves them just three points behind United.
There was controversy at Rugby Park as Partick Thistle secured a 2-1 victory over Kilmarnock to move level on points with the hosts in the battle to avoid a relegation play-off.
Vitalijs Maksimenko put Kilmarnock in front and, after Kallum Higginbotham had equalised, Michael Gardyne thought he had restored the advantage when his chip bounced down off the crossbar in a frantic start to the second half.
The home side were convinced that the ball had crossed the line before goalkeeper Paul Gallacher cleared, but referee Brian Colvin waved play on and Aaron Taylor-Sinclair rubbed salt in the wounds by scoring the winner five minutes from time.
Victory for Alan Archibald's side moved them above Ross County, who are now second from bottom after they were beaten 2-0 at relegated Hearts.
Jamie Hamill scored and was later sent off for a second bookable offence before Dale Carrick sealed the points for the Edinburgh club, who are on the verge of being taken over by local businesswomen Ann Budge.
Kenny McLean scored after less than 15 seconds and Paul McGowan was also on target as St Mirren secured a 2-0 home victory over Hibernian, despite playing with 10 men for an hour following Jim Goodwin's dismissal.
Both sides are three points clear of Ross County with four games remaining.
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