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The hosts were unable to find the try line in the first half, but four penalties from Freddie Burns left the Tigers 12-3 up at half time.
Goneva soon restored Leicester's advantage, scoring his first three minutes later, while Burns set him up for a second shortly after.
Falcons pressed hard to try to reduce the deficit, but Goneva intercepted the ball just five metres from his own line before running the length of the pitch to complete his hat-trick.
Andy Saull added a late consolation for the visitors, but they never looked like getting back into the match.