Fly-half Wilko kicked two conversions and a penalty to take his Premiership points total to 1,482 — two ahead of Andy Goode.
Newcastle director of rugby Steve Bates said: “It’s quite amazing he’s set the record for points in the Premiership considering how much rugby he has missed.
“We made a few errors and now go to Gloucester with confidence.”
Bristol head coach Richard Hill said: “There is nobody like Jonny when he’s playing behind a pack going forward. We contributed to our own downfall with far too many mistakes.”
Hill was right. The Falcons pack took control in the first half after Ed Barnes kicked a penalty and Wilko levelled.
The first try was vital, Adam Balding squeezed over and Wilkinson converted for a 10-3 lead in the 23rd minute.
Wilkinson sparked the game into life with a couple of mazy runs after the break.
And it worked as the Falcons went up a gear.
John Rudd crashed over in the corner after Wilkinson and Tom May carved a way through.
Wilko then converted and it was game over.