Stuart McCall's hosts trailed early due to a Stephen McManus own goal but after they had fought back through Henri Anier and Sutton, held on for the points despite Simon Ramsden's late dismissal.
The game was just 17 minutes old when the home fans were silenced - a header from Darren Barr, who had already denied Anier with a superb last-ditch tackle, deflecting off McManus and into the net.
The lead lasted just three minutes - an exquisite reverse ball from Keith Lasley sending Anier clean through and the Estonian striker retaining his cool to slot past Craig Samson.
There was another scare at the other end for the hosts moments later, when home goalkeeper Lee Hollis had to be at his best to prevent Barry Nicholson from restoring Killie's lead.
Iain Vigurs and Sutton put Samson into action but the visitors seemed to be heading for the half-time dressing room on level terms before some poor defending presented Sutton with a routine finish.
On-loan Celtic defender Jackson Irvine was the culprit - his wild attempted clearance finding fresh air and allowing Sutton the simplest of goals at the back post.
It was left to Samson to deny Sutton his second early in the second period while at the other end, Kilmarnock striker Kris Boyd produced two fierce drives to worry Hollis.
Ramsden's dismissal for a second bookable offence ensured a nervous ending during which Hollis had to smother danger on the goal-line on three occasions, but ultimately Kilmarnock failed to avoid a third straight defeat.