Fijian attacker Roy Krishna gave the Phoenix an early advantage before Jonatan Germano's thunderbolt established parity with Wellington going a man down in the 33rd minute through two yellow cards to defender Ben Sigmund.
Phoenix took the lead against the run of play through Tyler Boyd early in the second-half with David Williams' spot kick earning a point that Heart will be disappointed in after having most of the possession and chances.
After Heart had the first shot on goal through David Williams, Wellington took the lead in the sixth minute on the counter attack as Krishna stood up Patrick Kisnorbo, shifted the ball to the outside and fired underneath Heart goalkeeper Andrew Redmayne.
The home side had the ball in the back of the net five minutes later but Germano was correctly flagged for offside as he nodded Ben Garuccio's free kick past Glen Moss.
Heart continued to power forward and should have equalised on 15 minutes as Dugandzic blazed over after Williams' robbed the ball off Michael Boxall and found the winger unattended in the penalty area.
Despite their attacking endeavour, Heart were lucky not to have Kisnorbo sent off and a penalty awarded when the central defender appeared to handball in the 17th minute when blocking Krishna's goal bound strike.
The Phoenix eventually wilted just before the half-hour mark as Germano tied the match with a flush hit shot from outside the area after a cleared corner.
Within four minutes of the equaliser, Sigmund received two separate yellow cards for challenges on Williams and the Phoenix were reduced to 10 men.
With the man advantage, Heart did not make any inroads on the Phoenix goal before the half-time break as Redmayne was forced to make smart saves from Krishna and Boyd as the away side tried to strike on the counter.
Heart emerged from the sheds strongly and if it wasn't for a superb save from Moss, they would have taken the lead in the 49th minute as Dugandzic found himself one-on-one with the Phoenix goalkeeper.
The failure to hit the scoreboard hurt the home side on 56 minutes as the Phoenix took a shock lead when Krishna sprung a counter attack down the right.
Krishna's drilled cross was parried by Redmayne and Boyd was on hand to control and place the shot into the corner.
It only took the Heart six minutes to restore the deadlock as Williams made amends for an earlier miss by stroking home a penalty after Harry Kewell was brought down by Manny Muscat in the area.
The home crowd went wild in the 80th minute when Williams placed the ball past Moss but the goal was ruled out for a high boot by the attacker on Shaun Timmins; a decision which appeared to be marginal.
Despite pressing forward in huge numbers to grab the winner, Heart could not find a way past Moss with Williams coming closest deep into second-half stoppage time with a shot brilliantly blocked at point blank by the Phoenix custodian.
A draw for Heart means they are six points and significant goal difference adrift of sixth-placed Adelaide with four games to play, leaving a finals berth looking increasingly unlikely.
Wellington are now three points out of the final six and will need to win at least two of their remaining four games to have any chance of making the play-offs.
