But a last-gasp Rhys Priestland penalty snatched the points for Scarlets to secure their first away victory of the season and put them level on points with Connacht in the race for the top six.
Meanwhile Zebre remain bottom of the table on 15 points, and Bergamasco admits his team can no longer afford to dominate long passages of the game without capitalising on their chances.
"We had a good achievement on static phases giving good ball to our frontline," the flanker explained.
"We were dominant and also took measures to play against their strengths in defence.
"There were chances of winning today, and these must be seized if we are to grow - you do not grow, especially young players, with these honourable defeats that we are so tired of.
"As captain I'm happy with the attitude in 80 minutes, but less with our ability to make the most of our play and committing the mistakes that have brought chances to the opponents."
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