The Well striker pounced in the second minute after hesitation between Charlie Mulgrew and goalkeeper Fraser Forster to knock the ball into an empty net.
Celtic recovered to take control of the game but, at the start of the second half, after Hoops left-back Emilio Izaguirre had been adjudged by referee Euan Norris to have fouled Motherwell skipper Keith Lasley in the box, Sutton blasted in the spot-kick.
Although television pictures indicated the award was soft, Motherwell held onto the lead with all they could muster to register their first home win over Celtic since 2005.
On another day Celtic might have won with some ease but Motherwell, despite having to defend for long spells, could have added to their lead which will be a worry for Hoops manager Neil Lennon.
Celtic went into the game on top form, unbeaten in 17 games, but they got off to a nightmare start.
In the first minute Hoops defender Mulgrew missed a challenge on Chris Humphrey down the right, allowing the Well wide-man to bear down on Forster.
However, luckily for the visitors, defender Daniel Majstorovic was quickly over to block the shot at the expense of a corner which came to nothing.
But seconds later, Mulgrew and Forster both hesitated when Gavin Gunning sent a long, hopeful punt into the Celtic box and Sutton nipped in to round the keeper and gratefully slide the ball into an empty net.
Celtic's response was swift and moments later defender Mark Wilson climbed high at the back post to reach a Kris Commons free-kick but headed over the bar.
The energetic visitors were bossing a frantic match in which the ball was being battered from pillar to post.
Motherwell had half a chance when Jamie Murphy's cross from the left ended up at Humphrey but his angled shot from 14 yards was snatched and flew wide.
At the other end, with the home defence all over the place, Darren Randolph had to make a fine point-blank save from Anthony Stokes and seconds later the big keeper threw himself down to his left to push a Scott Brown shot from distance past the post.
Motherwell survived the corner but it looked only a matter of time before the Hoops levelled although Stokes again missed a chance in the 38th minute when he turned and drove high over from 16 yards.
In the final minute, as Celtic pressed again, the Irishman had a shot from 10 yards cleared away by Well defender Shaun Hutchinson as it sped past Randolph.
The start of the second half was as dramatic as the beginning of the first with Sutton grabbing his second goal from a penalty within three minutes.
Celtic failed to clear their lines after Hutchinson had headed a Forbes corner against the bar and, when Izaguirre tangled with Lasley in the box, referee Norris pointed to the spot and Sutton drove the ball past Forster.
Wilson and Stokes were taken off minutes later, replaced by Georgios Samaras and Efrain Juarez, and Celtic pounded the home side who were forced in to some frantic defending.
In the 54th minute Commons' long-distance free-kick beat the Motherwell defensive wall and Randolph but rebounded off the keeper's right-hand post.
It was another escape but, in a Motherwell breakaway, Murphy switched play to send Humphrey clear down the right only for the Well player to hesitate before getting his shot in and the ball struck Forster, who then pounced on the loose ball.
In the 65th minute James Forrest replaced Ki in what was Celtic's last throw of the dice with regards substitutions.
Celtic continued to push forward but with less conviction and in the 71st minute Motherwell carved open another chance on the break but Murphy's goal-bound shot from 12 yards was blocked by Sutton, who was trying to get out of the way.
The leaders kept coming and Mulgrew had a header from a Commons free-kick saved by Randolph and then Commons shot high over after finding a yard of space inside the Motherwell box.
In the first minute of added time Randolph tipped a Commons shot over the bar as it headed for the top corner as Motherwell defended stoutly until the final whistle before heralding their first home win over Celtic in six years.