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Billy Burns was unable to add the extra two points with a kick from touch, but a second Gloucester try five minutes later from Moriarty doubled the side's advantage. Burns converted, and the defending champions had an early 12-0 lead.
Burns added six more points with two penalty kicks before the half-hour mark, and a Steve McColl try on the stroke of half time following some good handling work from the Gloucester backs seemed to end the game as a contest.
Gloucester struggled after the break, and went down to 14 men when Lewis Ludlow was shown a yellow card.
Zebre made their opponents pay for their ill discipline moments later, when they were awarded a penalty try on 52 minutes. Edoardo Padovani converted, and added three more points with a penalty on the hour mark.
With Ludlow back on the field, the hosts saw out the second half to secure a 23-10 victory.