Sutton finished from close range in the fifth minute and Hearts were rarely troubled before Cammy Bell denied them a second when he saved Ryan Stevenson's 62nd-minute penalty.
But they had to survive some late pressure with Kris Boyd passing up two good chances for the hosts.
The victory did not move Hearts up the table but it did take them level on points with Hibernian, who host Aberdeen on Monday, and just two points off Kilmarnock, who slipped out of the top six in the dying seconds of the final pre-split fixtures.
Hearts made the perfect start, although they were given a helping hand by Mo Sissoko, who misjudged a diagonal ball.
Michael Ngoo decided the angle was too tight to shoot, but the on-loan Liverpool striker cut back inside Sissoko and showed the awareness to place a left-footed pass across the goalmouth which gave Sutton the simple task of tapping the ball home.
Hearts retained their edge and had the perfect chance to wrap up the game after Sissoko brought down Jamie Walker as the wide player ran into the box.
However, Bell stopped Stevenson's penalty and then denied Jason Holt from the rebound. Ryan O'Leary soon clumsily barged into Walker for what was arguably a stronger penalty claim but referee Murray waved play on.
The home side finally found their attacking feet in the final 20 minutes and put Hearts under serious pressure. Boyd had a free header from close range following a cross but his downward header bounced off the hard surface and over the crossbar.
Jamie MacDonald then pulled off an excellent stop from Boyd's penalty-box strike and both Sammy Clingan and Mark O'Hara fired just wide from close range as Hearts held on.