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Tries from Nikki Walker, Sonny Parker, Steve Tandy and Shane Williams, with three converted by Gavin Henson, earned Ospreys a maximum points haul at the Liberty Stadium.
Though the win was comfortable by the end, Ospreys were made to work for it, especially after Glasgow fly-half Colin Gregor had given the visitors a 6-0 lead.
Gregor added a third penalty after Nikki Walker had replied with a try for the Ospreys, but injured himself in the process and was replaced by Dan Parks.
These were the visitors’ last points as the Ospreys took control before the break thanks to tries to Sonny Parker and Steve Tandy, both converted by Gavin Henson.
The five-point victory was sealed in the second half with Shane Williams’s try.