Wilkinson kicked five penalties, two conversions and a drop goal as the Falcons registered their first win of the 2008/09 campaign.
The match was Wilkinson's first of the new season after undergoing shoulder surgery during the summer.
The operation was the latest in a long line of injuries to have struck the England fly-half since his last-gasp drop goal won England the World Cup five years ago but Wilkinson's impressive comeback will have reminded Ian McGeechan of his significant capabilities.
Elsewhere in the Guinness Premiership, 2009 Lions coach McGeechan had to watch his London Wasps side suffer their second consecutive defeat as Worcester Warriors ran out 11-10 winners at Adams Park.
On Saturday, Gloucester ended a 20-year wait for a league victory at the Rec as they downed Bath 22-17 to bounce back from their opening-day defeat to Leicester. The Tigers made it two wins from two, although a successful Peter Hewat conversion would have given London Irish a draw in their close fought encounter at Welford Road.
Saturday's other fixture saw Harlequins move to the top of the table tanks to a convincing 31-13 win over Bristol, who sit bottom of the table with no points from their opening two games.
On Friday night, 2005 Lions tourist Charlie Hodgson was the Sale Sharks hero as his late drop goal secured victory over winless Saracens at Edgeley Park.