Goals from Sutton and Zaine Francis-Angol either side of a Gunnar Nielsen penalty save from Hoops forward Kris Commons gave the home side a two-goal lead before Anthony Stokes reduced the deficit just before the interval.
Georgios Samaras levelled 10 minutes after the break before Celtic substitute Leigh Griffiths, on for Beram Kayal, grabbed a third in the 84th minute but as the game moved into injury time Sutton grabbed his second from a yard out to take the Steelmen to within three points of second-placed Aberdeen.
Both sides were applauded off the park and rightly so after a wonderful afternoon's entertainment although Hoops manager Neil Lennon will not be too happy about some of his side's defending.
The visitors, with John Herron making his first start as full-back, looked still to be asleep when the Steelmen took the lead.
A long punt by former Celtic skipper Stephen McManus evaded the lackadaisical Efe Ambrose and ended up at the feet of Sutton, who had time and space to set himself for an angled shot which, although not the most powerful, slipped under Hoops goalkeeper Fraser Forster for his 21st of the season.
The champions slowly asserted themselves and Motherwell dropped back towards their own penalty area but they were gifted a chance in the 13th minute when a slip by Celtic midfielder Kayal in the centre circle allowed Iain Vigurs to send Henri Anier clear.
The Estonian forward, however, could not shrug off the attentions of Ambrose and his shot was blocked for a corner which came to nothing.
Back came Celtic and they had the perfect chance to level when Vigurs barged Stokes in the box, and referee Craig Thomson pointed to the spot. But to roars of approval from the home fans, Nielsen dived low to his left to parry Commons' penalty away for a corner which came to nothing.
There were enough moments of slackness in the visitors' defence to encourage Motherwell when they did venture forward.
Celtic had the bulk of possession and pressure, though, and in the 37th minute it needed a fine interception by Motherwell left-back Stevie Hammell three yards from goal to prevent Stokes' cross reaching the unmarked Commons.
Two minutes from the break Celtic were punished for more ineptitude in their rearguard.
Hammell slung a cross from wide on the left into the box and when left-back Emilio Izaguirre and centre-back Virgil van Dijk failed to deal with it under pressure from Anier, Francis-Angol was able to slide the ball under Forster from six yards.
However, Motherwell could not get to the break without conceding.
With seconds remaining Stokes took a touch on a pass from Commons and curled the ball over Nielsen and into the far corner of the net to deflate the home fans and boost those in green and white.
The Parkhead men were out long before Motherwell for the start of the second half, indicating a sense of urgency.
There was no surprise that Celtic took control but a two-minute spell changed the course of the game
In the 53rd minute Fraser Kerr's header from Vigurs' corner was headed off the line by the perfectly-placed Kayal and two minutes later the scores were level when, from eight yards out, Samaras redirected Stokes' shot from the edge of the box past the helpless Nielsen.
The game sped from end to end with chances for both sides, Forster making a save at his near post from Anier before there was a penalty claim from Motherwell fans who thought Samaras had handled the ball in the box.
With six minutes remaining Griffiths sneaked in behind the Well defence to slam the ball past Nielsen and it looked enough to give the visitors all three points, but with seconds remaining Sutton bundled the ball over the line from close range after being set up by substitute Lionel Ainsworth for a deserved point.