A Jared Payne hat-trick and a brace from Tommy Bowe catapulted Ulster to their second win on the bounce, and they now need just one more victory from their remaining games to guarantee qualification for the Champions Cup.
And after seeing his side impress in Parma at the weekend, Kiss is keen to see the fine form continue to seal a play-off semi-final berth.
"It is mission accomplished," said Kiss. "Five points was what we expected from ourselves and it's what was expected of us, so it's good to deliver that.
"We've put together a good couple of games, with beating Connacht and now Zebre.
"We're in the top four, and it's our job to stay there."
Ulster face top-of-the-table Leinster at the Kingspan Stadium on April 30, a must-win game for the visitors if they are to hold off Connacht in the battle for first place.
And with tough ties on the agenda with a trip to Ospreys following next week's Leinster visit, Kiss admits there is no room for complacency.
"We've got a good preparation week building for what is a classic in our calendar," he said.
"Leinster did well to get their win over Edinburgh. It's not going to be easy, we know that.
"At Kingspan last year we did a job on them and it's up to us to do the same again.
"There's no doubt it's going to be a massive atmosphere and our boys will be up for it."
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Guinness PRO12 Ulster News
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