The fly-half finished the match with a personal points tally of 17 points, including a well-taken try in the closing moments of the match.
This win will help relieve the pressure on the Exiles, who now have three Premiership victories under their belt as they attempt to consolidate.
Leicester, hit by international call-ups, had remained competitive throughout the match with a try from Ollie Smith and a conversion from Steve Booth putting them on the scoreboard.
But Everitt's four penalties kept the Exiles ahead, before the home side cut loose in injury time, Everitt and then Michael Horak crashing over for two late tries.
Scorers:Tries: Horak, Everitt. Cons: Everitt. Pens: Everitt (4). Drop Goals: Everitt.
Leicester:Tries: Smith.Cons: Booth.