Kilmarnock 0-1 Hearts
John Sutton's early tap-in gave Hearts a narrow but deserved win over Kilmarnock, who rallied late on at Rugby Park.
Michael Ngoo provided Sutton an easy finish as the hosts started slowly.
Killie goalkeeper Cammy Bell saved Ryan Stevenson's second-half penalty after Mo Sissoko felled Jamie Walker.
Kris Boyd twice nearly rescued a point for the home side but headed over one chance before Jamie MacDonald smothered another.
Hearts, who now move level on points with Hibs in the Scottish Premier League, will have encouraged their support with this energetic and forward-thinking performance.
Kenny Shiels, meanwhile, will have to ensure his Kilmarnock charges get their teeth into future games quicker than they did here.
Killie are actively trying to attract as many fans as they possibly can to the closing matches of this campaign, but their start to this encounter will not have acted as a catalyst to that process.
It seemed all too easy for Hearts' tall forward Ngoo to collect Stevenson's long-ball and calmly cut inside Killie's Sissoko to square for Sutton's simple eighth goal of the season.
The home side's shakiness continued as defender Ryan O'Leary and Rangers-bound goalkeeper Bell combined to make a hash of a clearance, then Sutton blasted over after another Ngoo set-up.
Killie's two new Nigerian recruits; Gabriel Reuben and Papa Idris, watched on in the stands as their countryman Rabiu Ibrahim scurried around in vain trying to create openings for Paul Heffernan and Boyd up top.
A glanced header wide by Hearts midfielder Walker was followed by something for the Rugby Park home crowd to finally get an inch off their seats about. Killie's Dutch left-back did well to find Heffernan in the box, but a sprawling Andy Webster blocked the Irish striker's effort.
In addition to impressive link-up from Sutton and Ngoo, Walker and Jason Holt were a regular scourge to Shiels' team.
James Fowler's deflected effort just before half-time was not enough to prevent booing from Kilmarnock fans as the teams left the pitch to take stock.
Two half-time substitutions perhaps reflected Shiels' unhappiness, while Dale Carrick had an early sitter after entering the fray for the visitors.
Home fans starved of chances to applaud dished out generous loud claps when Chris Johnston's penetrating run ended with a shot on target, albeit comfortably gathered by MacDonald.
Walker picked up where he left off for Hearts, surging into the box to draw a clumsy foul from Sissoko and referee Calum Murray pointed to the spot. However, Bell saved Stevenson's kick, and Holt's follow-up, expertly.
And Hearts should have had another penalty almost immediately, O'Leary the next to misjudge how to thwart Walker, but this time nothing was given.
With Hearts desperate to close-out the match, superb wing-play on the left by sub Arvydas Novikovas led to Dale Carrick thumping narrowly wide.
Despite dominating, the final say almost went to Kilmarnock.
Johnston's cross was headed over by Boyd, who then forced MacDonald into a crucial close-range save minutes later.
And there was also time for Mark O'Hara to drill a foot wide as the home side finished how they wish they had begun.