Goals in either half from Josip Tadic and Richard Garcia proved the difference although the margin could have been ever greater had Heart taken full advantage of their numerous chances.
David Williams, a constant threat down the left all night, and Fred missed a couple of clear opportunities for Heart, while Adelaide goalkeeper Eugene Galekovic made a string of stops.
But it mattered little as Heart picked up their sixth win of the campaign to move up to fifth and boost their chances of appearing in the finals for a second consecutive season.
Adelaide remain in second place and cannot be moved from that spot by the end of the weekend.
Williams had the first sight on goal in the second minute, but his speculative effort from just outside the box flew over the crossbar without really troubling Galekovic.
Just three minutes later, the home side took the lead when Fred and Williams worked a short corner cleverly and the Heart skipper floated a superb ball across the ball-watching Adelaide defence for Tadic to nod home for his sixth of the season.
Heart continued to dominate possession in the period following the goal, but the visitors did fire a couple of warnings at the other end to keep them on their toes.
Andrew Redmayne was forced to palm away a Dario Vidosic cross with Jeronimo bearing down on goal, while Daniel Bowles failed to find the target when he shot wide with the follow-up.
But their efforts counted for little as Garcia and Williams in particular continued to cause the visitors problems in the attacking third.
Williams should have doubled Heart's advantage just moments before half-time, but Galekovic was good enough to parry away his effort and Tadic failed to connect with the loose ball.
Vidosic and Cameron Watson could only shoot over when they tried their luck from outside the box late in the half for the visitors, but Heart fully deserved their one-goal lead at the break.
Heart fans had barely settled back into their seats when they had the second goal of the match and it came through the ever-reliable source of Garcia.
Matt Thompson used his bulk to hold off the Adelaide defence, before finding Garcia and the veteran fired past Galekovic for his fifth goal of the season.
Heart had a scare when Fred limped off just minutes later with yet another hamstring problem, but the hosts were in cruise control and Williams could have even made it three when Galekovic sprinted out to leave the goal unguarded just on the hour mark.
The Heart striker failed to take advantage of the situation and the game then became a bit scrappy with the guile of Fred and class of Vidosic - who was also substituted off after he picked up a knock - missing from the middle of the park.
Then with 15 minutes remaining Heart began to seize up somewhat as Adelaide sensed an opportunity with the recent late vulnerability of the home side.
Jon McKain nearly pulled one back for Adelaide from a well-worked set-piece moments after coming on as a substitute, but that was as good as it got for the visitors as Heart held on for a well-deserved triumph.