Wilkinson had previously held the record until Leicester's Andy Goode moved to the front during last season. Goode scored a total of 1,480 points during his time with Tigers and Saracens but has since departed to French club Brive.
Wilkinson's touchline conversion of John Rudd's 58th-minute try moved the England fly-half two points clear of Goode's tally as he turned in another dominant display for the Falcons.
"When Jonny kicked that conversion from the touchline in the second half it was just what we needed," said Newcastle director of rugby Steve Bates.
"It gave the team a big lift but that's what you get with Jonny. It's quite amazing that he has the record for points scoring in the Premiership considering how much rugby he has missed."
Those sentiments were echoed by Bristol supremo Hill who emphasised the impact Wilkinson had on the outcome of the match.
"There's no-one like Jonny when he's playing behind a pack going forward and we contributed to our own downfall by making far too many mistakes."