Sutton struck early in the second half to cancel out Melvin De Leeuw's 30th-minute opener before Rob McHugh put the result beyond the visitors late on.
The former Hearts striker tested visiting keeper Mark Brown early on and Scotland international James McFadden was a constant threat as Well took the game to the Highlanders in the opening stages.
However, they fell behind in the 30th minute when a shot from De Leeuw took a wicked deflection and flew in off the post beyond keeper Lee Hollis.
County, who had rarely threatened the home goal, had a great chance to double their lead before half time but De Leeuw failed to keep his effort down after fine work by captain Richard Brittain.
And he was made to pay when Sutton (54) headed home from a corner after Brown had pulled off a stunning save to beat the same player's free-kick round the post.
Hollis was called into action again to keep out a goalbound Brittain 25-yarder before Sutton (68) latched onto Henri Anier's pinpoint through ball to slip it beyond the reach of Brown.
And there was no way back for County when substitute McHugh headed home the killer third for Motherwell with 13 minutes remaining.