Pollard conceded 17 runs in two wicket-less overs, as the Stars successfully chased 153 for victory in the 22nd match of the tournament at the Melbourne Cricket Ground.
Discarded Australia fast bowler Ben Hilfenhaus notched the Stars over the finish line, when chipped a delivery from medium-fast bowler Michael Neser through square leg for a single with three deliveries remaining.
Earlier, Pollard struck a couple of boundaries in 16 from 13 balls before Hilfenhaus had him caught at mid-on in the penultimate over from a miscued pull, as the Strikers reached 152 for eight from their allocation of 20 overs, after they were put in to bat.
The result meant that the Stars now have six points � the same number of points as Perth Scorchers � but they are third in the standings, due to a superior net run rate.
The Strikers, whose squad also includes the Barbados-born, England all-rounder Christopher Jordan, are one of four teams with four points. They are however, sixth in the standings, due to net run rate.
Brisbane Heat, whose squad includes West Indies Twenty20 star leg-spinner Samuel Badree, and Sydney Sixers are the current front-runners with eight points each � but the Heat have the superior net run rate.
The BBL continues on Wednesday, when Badree and the Heat face the Scorchers at the Gabba, while Pollard and the Strikers have to wait until next Monday to see further action when they host the Melbourne Renegades, whose line-up includes star West Indies spinner Sunil Narine, at the Adelaide Oval.