Tasmania lost their opening two matches of the tournament, but Rose smacked 26 runs from nine balls on Tuesday to make it three wins on the trot.
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The dramatic finale overshadowed Harris' first hit-out with the Bulls since he starred in the Ashes, with the Australian Test fast bowler returning from a hamstring injury to take 3-61 off 9.1 overs.
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Evan Gulbis was the pick of the bowlers taking 4-36 off his ten overs, with the Tasmania attack forced to make do without the services of key paceman Ben Hilfenhaus who returned home for the birth of his first child.
The win lifts fourth-placed Tasmania to eight points, one shy of Queensland in second.