The New Zealand side's recent run of good form came to an abrupt halt as the team in red and white ran amok at the 'Cake Tin'
David Williams' hat-trick, together with goals from Dutch duo Orlando Engelaar and Robbie Wielaert, made it three victories in successions for Heart, who have now won four of their last five games and are unbeaten since January 10.
The Phoenix had been one of the form teams in the A-League going into the clash at Westpac Stadium, but they had no answer to the inspired visitors.
Heart remain bottom of the table but are only seven points off sixth-placed Sydney and by no means out of finals contention, a notion that would have been inconceivable following their nightmare start to the 2013-14 season.
Having recorded their first genuine away win since December 2011 in the most emphatic fashion imaginable, the biggest test yet for Van 't Schip's side will arrive with the visit of league leaders Brisbane Roar to AAMI Park next Sunday.
Somewhat surprisingly, Phoenix coach Ernie Merrick made four changes to a winning team, with Michael Boxall, Jeremy Brockie, Kenny Cunningham and Jason Hicks, a goal-scorer off the bench in a 3-2 triumph away to Newcastle Jets last weekend, all coming into the team. Tyler Boyd, Matthew Ridenton, Josh Brindell-South and Reece Caira were the players to make way.
Williams, a match-winning substitute in Heart's previous two fixtures, replaced the injured Harry Kewell in the visitors' starting XI. Patrick Gerhardt and Engelaar, available after serving a suspension, came into midfield for Jonatan Germano and Stefan Mauk.
Heart's strong start to the game saw Engelaar open the scoring in the ninth minute, the marquee international powering home a firm strike from just outside the area after Wellington failed to clear their lines.
The Phoenix conspired to gift the away side a second goal 10 minutes later, when goalkeeper Glen Moss charged off his line in an ill-judged attempt to intercept a quick free-kick.
Moss found himself stranded in no-man's land and Williams, still with plenty to do, was able to direct a well-taken finish into an empty net.
The Phoenix had scored 15 goals in their last six matches going into this game, but there was no sign of that attacking fluency in the first 45 minutes, with Carlos Hernandez's set-pieces providing their only threat.
The game was over as a contest five minutes into the second half, when Massimo Murdocca won possession near the half-way line, springing a fast counterattack and exchanging passes with Williams, who capped off the move by slamming a cool finish underneath Moss.
The onslaught continued four minutes later, Wielaert smacking a shot against the crossbar which rebounded off the back of the unfortunate Moss and into the net for 4-0.
And a few minutes past the hour mark, Williams completed his hat-trick, curling a brilliant long-range strike to the far post and beyond the reach of the beleaguered Wellington goalkeeper.
The thrashing was a rude interruption to the previously resurgent Phoenix, who had playmaker Hernandez comprehensively shackled by Gerhardt and suffered as a consequence.
Having slipped outside the top six, they must now prepare themselves for a difficult game away to reigning champions Central Coast on Friday night.
Melbourne Heart 5 (Engelaar 9', Williams 19', 51', 63', Wielaert 54')
Wellington Phoenix 0