After Goode added a penalty to stretch his side's lead to 16-9, the Falcons levelled with Robbie Morris' late converted try as Wilkinson came through the entire 80 minutes.
Charlie Hodgson emerged triumphant in the battle of the prospective England kickers as he scored all of Sale's points in their 18-10 victory over Wasps at Edgeley Park to take them to the top of the table.
The match had been billed as the "game of the season" by Sharks head coach Philippe Saint-Andre - and it did not fail to disappoint. But despite there being a clutch of world-class backs on show, it was a contest dominated by the forwards and as a result, it all came down to the kicking.
Hodgson, whose role as kicker for England could be challenged by the emergence of Wasps' New Zealand-born full-back Mark van Gisbergen, planted over six penalties. Van Gisbergen replied with one penalty and a conversion, but Wasps were overrun by a dominant Sharks pack.
Rock-bottom Leeds slumped to their fifth successive defeat, 22-15, after being put through the mincer by Worcester's forwards at Sixways.
Skipper Justin Marshall worked tirelessly to try to lift Leeds out of the doldrums, but they never seriously threatened to score a try, and Italian fly-half Roland De Marigny booted all their points through an immaculate goalkicking display.
Worcester, in contrast, preserved their unbeaten home league record this term as wing Thinus Delport and replacement flanker Saosi Vaili scored tries, with fly-half Shane Drahm adding four penalties.