The Reds started well against a scrambling Victory defence but it was the visitors who struck first through in-form Fahid Ben Khalfallah with Melbourne's first genuine opportunity in the ninth minute.
Pablo Sanchez equalised just prior to half-time before the Reds went ahead when Dylan McGowan converted a fortuitous header on a second-chance opportunity after he had been off the field.
But United's lead was short-lived as long-timer Adelaide tormentor Thompson scored three minutes later to ensure a split of the points between the two arch rivals.
Though Victory missed the chance to move top of the table, Thompson's late effort ensured they jumped up to second spot.
In Saturday's other game, Newcastle Jets won their first away match of the season to jump off the bottom of the A-League with a 2-1 win over Western Sydney Wanderers.
First-half goals to Enver Alivodic and Lee Ki-Je had the Jets in complete control at the break before a sublime free-kick from Nick Kalmar six minutes into the second half got the home side back in it.
Wanderers could not find an equaliser, though, and replace the Jets at the foot of the table.
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