Tigers flanker Salvi topped the fans' vote as their Player of the Year.
Salvi, who has also been nominated for the Aviva Premiership Player of the Season, said: "It's a great honour and very humbling.
"I've been here nearly two years now and I've enjoyed it immensely and I feel like part of the family. I'm very honoured to pick up the award.
"I am happy with how things are going and when you are happy in an environment like this you can play good rugby.
"The supporters at Tigers are phenomenal. They get behind us and back us every week and to have them voting for me is humbling."
Manu Tuilagi, who earlier in the day celebrated his call-up to the British & Irish Lions tour squad, enjoyed an awards double as he was the supporters' choice for Young Player of the Year for a second time in three seasons and also picked up Try of the Season for his score against Worcester Warriors at Welford Road.
Tuilagi said: "It's a good feeling to win two awards. You work hard in training and through the season and to get something like the Young Player of the Year is amazing.
"The try was a nice team try. All of the hard work was done by the team and I only had to go the last five metres or so."
Adam Thompstone, with nine tries to his credit so far in his first season with the club, was named Newcomer of the Year.
Thompstone said: "It was a surprise - you don't really expect to win an award like this. Credit to the boys, they have been really welcoming and have made it easy for me to settle in."
Martin Castrogiovanni was the supporters' choice for the Outstanding Service Award and is the recipient of a Tag Heuer watch courtesy of sponsors Goldsmiths. The popular prop has made almost 140 appearances in seven seasons with the club and is approaching a century of caps for Italy.
Hooker Tom Youngs last month collected the Players' Player of the Year award after topping the votes of his Tigers colleagues. Prop Fraser Balmain was the Players' Young Player of the Year.