Daniel Kickert's lay-up for Melbourne forced overtime at 71-71, but Shawn Redhage's four-point play with one minute and 42 seconds left in the extra period saw the Wildcats take control, with the reigning NBL champions eventually winning by a point.
The result saw Perth move to an 11-5 win-loss record in the Australian league - third in the standings and two wins clear of fourth-placed Melbourne (9-8).
United led 22-21 in front of their home crowd after the first quarter but the visitors were better across the middle two periods to hold a 53-47 lead at three-quarter time.
The home side reeled Perth in, however, to ensure an extra five minutes would be needed, although Melbourne could not get the job done, suffering their second loss to the Wildcats in three meetings this season.
DeAndre Daniels was critical to the Wildcats' victory, scoring 18 points and claiming 13 rebounds.
The top three teams all won in round 12 with leaders the New Zealand Breakers overcoming the Sydney Kings 97-85, while the second-placed Cairns Taipans defeated the Wollongong Hawks 100-93.
Tai Wesley top-scored for the Breakers, despite starting on the bench, sinking 23 points, while Stephen Weigh scored 21 points for the Taipans.
New Zealand lead the NBL standings with a 13-5 record, while Cairns (12-5) have a game in hand.
In the other match, the Adelaide 36ers triumphed 91-84 over the Townsville Crocodiles.