Everitt, who replaced the injured Mark Mapletoft in the Exiles' starting XV, took advantage of an ill-disciplined Sharks outfit – slotting over six penalties to help clinch the win.
The Exiles No.10 opened the scoring with a penalty in the ninth minute, but the Sharks claimed the lead three minutes later when Graham Bond charged down an attempted clearance kick by Michael Horak, leaving the tryline open. Braam van Straaten converted to stretch the lead.
Three more successful penalty kicks by Everitt and a try for centre Rob Hoadley, however, gave the Exiles a 19-13 lead at the break.
The second half began with Everitt and Van Straaten trading penalties, before Sharks winger Steve Hanley – a former England player – went over for a try late in the half, taking the lead with four minutes to spare.
Everitt's drop-goal in stoppage-time, however, handed the Sharks their fourth home defeat of the season.
For Sale:Tries: Bond, HanleyCons: Van Straaten 2Pens: Van Straaten 4
For London Irish:Try: HardwickCon: EverittPens: Everitt 6DG: Everitt