Motherwell were rewarded for their dominance five minutes before the break when Sutton got in behind the Hearts defence on to a curling Iain Vigurs pass and, after touching the ball past the advancing Jamie MacDonald the striker struck from a tight angle, with the ball crossing the line off Brad McKay.
Hearts goalkeeper MacDonald made an instinctive block after just three minutes when James McFadden's deep cross was almost turned into his own net by Jordan McGhee.
And the home team survived another scare when McFadden exchanged passes with Steven Hamill and set up Lionel Ainsworth in front of goal, but his net-bound shot was blocked by a last-ditch challenge from Scott Robinson.
The Hearts keeper then got down to his right to turn an Ainsworth volley wide before defender McKay blocked a Sutton drive when MacDonald misjudged a delivery into the penalty area two minutes before the opening goal.
Vigurs opened up the Hearts defence again six minutes after the break to put Ainsworth through but MacDonald got down at the feet of the Motherwell forward to make a brave bock.
Hearts came close to an equaliser five minutes later when Motherwell keeper Gunnar Nielsen almost fumbled a powerful drive from Smith.
Jamie Walker then sent a half-volley narrowly wide as the home side grew in confidence and the Hearts substitute forced Nielsen to parry another drive.
Motherwell held on to see the game out though and secure a sixth successive top flight win for the first time in 77 years.