The New Zealand outfit made it two wins in succession after getting off the mark the previous week, while John Aloisi could well be facing the axe
Stein Huysegems scored the only goal of the game with 15 minutes to play as the Heart fans showed their discontent to their coach by chant in the dying stages.
The Heart settled first early and probed to find an opening but the Phoenix still looked dangerous in attack with Carlos Hernandez finding space in front of the home defence.
After a couple of early David Williams' sights on goal, the Phoenix could have taken the lead with two clear opportunities before the 10th minute.
Jeremy Brockie found himself one-on-one with Heart keeper Andrew Redmayne after Manny Muscat's long ball but screwed his shot well wide of the right post.
Hernandez looked likely to make something happen with the ball at his feet facing the Heart defence and after Albert Riera's superb defensive work to win back possession, the Costa Rican's shimmy and low shot slammed off the bottom of the upright.
Heart had the best chance of the first half with Jason Hoffman releasing Michael Mifsud down the right channel and his square to Williams at the back post was somehow side-footed wide with plenty of the goal to aim at.
With both sides desperate to get a win, the game was always going to open up further in the second-half, as John Aloisi gave full back's Aziz Behich and Hoffman a free reign to get forward.
The move immediately looked to unsettle Phoenix with Behich making two slaloming runs through the away defence with one shot saved by keeper Glenn Moss and the other chance well-snuffed out by Louis Fenton.
Despite sending in plenty of balls into the penalty area, Heart couldn't apply the finishing touch with Mifsud striking Moss from point-blank and Williams lashing a volley over the bar from inside the six-yard box.
Captain Harry Kewell had a quiet night, playing the full 90 minutes for the first time since his return from injury, and he nearly ignited the home fans with a curling effort from the edge of the box going inches wide.
The home side pressed forward searching for that elusive goal and they were given several warning signs with Brockie firing high from a tight angle and also putting a free-header over the bar.
Eventually one of the Phoenix counter-attacks found the target in the 75th minute as substitute Tyler Boyd eluded Behich and slid a cross into the six-yard box which Huysegems duly stabbed home.
Aloisi threw Golgol Mebrahtu on as the Heart desperately searched for the equaliser but the Phoenix defended deep and compactly to not allow their opposition a chance to share the points.
The home fans chanted 'Aloisi out' and 'you are getting sacked in the morning' as the Heart crashed to their 17th game without a win.
Melbourne Heart 0
Wellington Phoenix 1 (Huysegems 75')
Attendance @ AAMI Park: 6,486
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