"To be honest, I don't know what to say," said Goneva who has just agreed a new contract at the club. "I am really happy and I thank the team for the award tonight."
Fly-half Williams also received a Tag Heuer watch after being named Players' Young Player of the Year in his first season with the club.
"I'm proud and honoured the boys have voted for me," Williams, 22, said. "I'm really grateful. When I signed I didn't think the season would go as well as it has for me."
The presentations came at the end of an evening in the company of player sponsors, 1880 Club members and guests in the Premiership Suite at Welford Road.
As part of their sponsorship of the club, Savile Row tailors Chester Barrie will prepare a made-to-measure suit for Niki Goneva's player sponsor from Posinstall.
The Tigers Supporters' Awards Night is at Welford Road on Tuesday, April 29. Voting is open to all supporters in four categories until the end of the Easter weekend. Click here for more details of how to vote - and nominate a Fan of the Year.
Tickets for the evening, featuring a three-course meal with the Tigers squad, videos and interviews, plus the award presentations are on sale now, price £45 season ticket holders, £55 all other supporters. Click here to book your place.