Sutton scored his eighth goal of the season from close range after he got on the end of an inviting cross from Michael Ngoo, who had latched onto a long ball forward from Ryan Stevenson.
Hearts then spurned the opportunity to double their lead after 62 minutes when they were awarded a penalty after Jamie Walker was pulled down in the area by Mahamadou Sissoko.
Killie keeper Cammy Bell saved Stevenson's spot-kick low to his left before denying Jason Holt as he followed up.
Sissoko was also at fault for the Jambos' goal as he misjudged the ball from Stephenson that led to the goal and another mistake from the French centre-back almost led to another.
His weak clearance allowed Jamie Hamill to play the ball into the box but Jamie Walker could only glance his header wide of the mark.
The home side struggled to get into the game as Hearts continued to dominate, despite Stevenson's missed penalty.
Another strong penalty shout for Gary Locke's side soon afterwards was denied by referee Calum Murray, who waved away the protests after Ryan O'Leary barged into Walker in the area.
Hearts substitute Arvydas Novikovas then put in an inviting cross for Dale Carrick but he could only put the ball over the bar after beating Bell to the ball.
Killie finally showed in the final 20 minutes of the contest, with Kris Boyd twice missing out.
First the striker saw his close-range header bounce into the turf and over before Hearts keeper Jamie MacDonald kept out his shot from the edge of the penalty-box.
Samy Clingan and Mark O'Hara also went close late on but Hearts held on to secure maximum points.