Veainu and Greg Bateman sealed a bonus-point for the Tigers with a James Ambrosini 39th-minute penalty all Treviso had to show for their efforts.
The Italians find themselves bottom of Pool 4 after two games having succumbed 32-7 last weekend to GUINNESS PRO12 rivals Munster.
They have now lost 16 consecutive games in this competition and Goosen said: "Leicester had players with the quality and ability to beat players and score tries, even in poor weather conditions, and that was the difference between the teams.
"There were times when we matched Leicester but in those conditions you have to turn territory into points and we didn't take our opportunities."
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