Adelaide had lost one of the previous 13 A-League meetings between the teams and knew a win on ANZAC Day would move it to the top of the table, above both second-placed Central Coast Mariners and leader Melbourne City.
It appeared set to do just that when Stefan Mauk converted a cut-back cross following a run by Ryan Strain down United's right flank, breaking the deadlock in the 51st minute.
However, that lead did not last too long. Ben Waine made it four goals in as many games to get Wellington level, seizing on a loose ball before rounding goalkeeper Joe Gauci to score.
The contest at WIN Stadium appeared set to finish as a draw, only for Wellington to have a chance to win it late when Gauci was adjudged to have fouled Phoenix substitute Jaushua Sotirio during a scramble inside the penalty area.
Hemed held his nerve to convert the penalty in the 97th minute, clinching Wellington – who are on a four-game unbeaten run - a first league triumph over Adelaide since December 2019.
Brisbane Roar had also been due to host Perth Glory on Sunday, but its fixture was postponed following the introduction of a three-day lockdown by the Western Australian and Queensland Governments in relation to the Perth and Peel areas.