Wales hooker Ken Owens scored the only try of the game as Cardiff slumped to a 16-6 loss at Parc y Scarlets.
And after a tough battle with opposite number James Davies in Llanelli, Warburton is determined to produce a dominant performance against the Italians.
"I want to get a win at home against Zebre and give the fans something to cheer - it is that simple," he told the Cardiff Blues website. "We owe them a performance on the final day.
"People talk about motivation at this time of the season, but I never find it difficult to play for Cardiff Blues.
"I'm very proud to represent this region. Every time I put this jersey on I want to do it justice.
"I love playing for the Blues. I was given the option to take time off in the latter stages of the season but I want to be involved every game and do everything I can."
Meanwhile the Blues have announced their intention to hold interviews for the vacant head coach role at the end of the season.
And Warburton admits after a season of unrest, with Dale McIntosh and Paul John taking temporary control since Mark Hammett left in February, the one thing the club needs is stability.
"We have got to look to the future now," he added.
"We all realise that we need some stability on the coaching front and whoever takes the top job will get at least three years to build. That is crucial and pleasing to hear as we look ahead.
"We've got to be realistic and realise the building process starts now. We have got to improve year on year and I hope our fans will get right behind us on this journey."
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