The loss leaves County bottom of the table while earning the visitors their first away points of the campaign.
Motherwell broke the deadlock four minutes into the second half thanks to a brilliant free-kick from Vigurs which flew into the top corner.
And six minutes later it was 2-0 as Lee Erwin found Sutton who, from 35 yards, lofted the ball over keeper Antonio Reguero for another excellent strike.
However, County reduced the arrears 12 minutes from time thanks to Liam Boyce who turned the ball home from close range.
But despite plenty of home pressure during the final stages – Darren Barr and Yoann Arquin both had chances – it was another setback for County.
In a goalless first half, Melvin De Leeuw and Michael Gardyne went close for more dominant County on more than one occasion, while Erwin had a header cleared off the line for the visitors.