Motherwell started well but fell behind on the half-hour mark when Melvin de Leeuw's heavily deflected shot went in off the post.
But Sutton headed into the top corner from a corner-kick in the 54th minute and then finished brilliantly into the other side midway through the second half.
And substitute Bob McHugh provided another excellent finish to secure a 3-1 Scottish Premiership victory for the home side at Fir Park.
It was the first time Motherwell have scored more than twice in a game this season and they showed their attacking intent by creating three chances in the opening five minutes.
Mark Brown parried Henri Anier's shot after the Estonian had linked up well with Sutton and Shaun Hutchinson headed over from the resulting corner.
Zaine Francis-Angol then shot over from the edge of the box after good work from James McFadden and Anier.
Graham Carey sent in some dangerous crosses from Ross County's left flank but Motherwell continued to edge play, although it took until the 25-minute mark for them to create another chance.
Some smart play from McFadden and Carswell handed Sutton a shooting chance but Brown saved with his foot and Anier flashed the rebound past from 20 yards.
The visitors took the lead with another move down the left. Carey held the ball up and released the overlapping Ben Gordon, whose cutback found de Leeuw. The Dutchman, who came off the bench against Hearts the previous week to score a late equaliser and set up the winner, kept his shot low and it took a wicked deflection before flashing in off the post.
County soon lost Brian McLean, who had been injured in an accidental collision with Sutton, with Grant Munro coming on.
Motherwell pushed to get back on level terms and Sutton, Simon Ramsden and Hutchinson all got on the end of McFadden set-pieces, but none of them could seriously trouble Brown.
And County had a better chance before the break when de Leeuw cut the ball back for Brittain. The County captain had time and space 16 yards out but could not keep his shot down.
Motherwell manager Stuart McCall, who had an unnamed injury doubt leading into the game, brought on Lionel Ainsworth for McFadden at half-time.
De Leeuw had another chance when he hooked Brittain's deep cross towards goal but Lee Holis comfortably held.
Sutton won the corner that he would equalise from when his curling free-kick was pushed just wide. Ainsworth delivered and Sutton rose well to power a header towards the top corner from 10 yards out. Brown did well to get a hand to it but could not prevent the ball crossing the line.
Motherwell kept on the pressure but Hollis had to react quickly to save Brittain's deflected long-range strike before Anier sent a diving header just wide from Ainsworth's cross.
And County hit back well with Stuart Kettlewell forcing a good stop from Hollis following a smart move.
Sutton's second goal came when County were caught square at the back after Motherwell won a loose ball near the halfway line. Anier played an instant pass over the top and Sutton ran clear before guiding the ball high into the net for his fifth goal of the season.
The assist proved Anier's last action before being replaced by McHugh.
Motherwell built on their lead and Ainsworth had a shot saved on the break before McHugh turned and shot over from a difficult chance.
McHugh soon found his range in the 77th minute when he ran on to Ramsden's diagonal ball and bulleted a header into the top corner from 12 yards.
Motherwell kept pressing and Sutton twice came close to his hat-trick with headers while substitute Iain Vigurs had a long-range drive deflected just wide.