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.
Hearts manager Gary Locke stressed on Thursday that his team still had much to play for with about #300,000 more prize money available if they moved up from 10th place to seventh.
The club's tight finances were later underlined when Darren Barr and Gordon Smith were told they would not be offered new deals, although the former returned from injury to start the game.
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.
Locke's players heeded his words with a perfect start, although they were given a helping hand by Mo Sissoko, who misjudged Stevenson's 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.
Centre-back Sissoko almost got his team into trouble again when his weak clearance allowed Jamie Hamill to lob the ball into the six-yard box with Jamie Walker glancing a header past the far post.
Kilmarnock handed Rabiu Ibrahim his first start since his January move from Celtic and the midfielder was at the heart of one good passing move, but that was a rarity in a disjointed first-half display from the hosts.
Walker had a couple of decent long-range efforts for Hearts, who comfortably dealt with Kilmarnock's attacks, although they were grateful to see James Fowler's shot spin wide off the back of Andy Webster.
Kilmarnock manager Kenny Shiels made some major changes at half-time with Ibrahim and Jeroen Tesselaar going off and Chris Johnston and Borja Perez coming on. Mark O'Hara dropped back as Killie changed from 4-4-2 to a 3-5-2.
Hearts were soon forced into a change when Ngoo went off with a shoulder injury as Dale Carrick came on.
Johnston gave the home fans in the 3,572 crowd something to cheer with a run and shot, although Jamie MacDonald saved comfortably. Hearts retained their edge at this point and had the perfect chance to wrap up the game after Sissoko brought down 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.
Hearts had another good chance when substitute Arvydas Novikovas embarked on a strong run and delivered an excellent cross into the danger area. Carrick got there just before Bell but could only knock the ball just over the bar.
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 Johnston's cross but his downward header bounced off the hard surface and over the crossbar.
MacDonald then pulled off an excellent stop from Boyd's penalty-box strike and both Sammy Clingan and O'Hara fired just wide from close range as Hearts held on.
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- Ryan Stevenson