Bates said: "When Jonny kicked that conversion from the touchline in the second half it was just what we needed.
"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.
"We're very pleased with the result although we did make a few errors and we now go to Gloucester with confidence.
"We will keep our feet on the floor because while it was a good performance, it wasn't a vintage performance."
Bristol head coach Richard Hill also paid tribute to Wilkinson, adding: "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.
"I can't put my finger on why that happened and I'm at a bit of loss really after we played so well last week."
The Falcons pack took control in the first half after Ed Barnes kicked a third-minute penalty but Wilkinson levelled the scores in the eighth minute.
Adam Balding then blasted over for the opening try and Wilkinson converted for a 10-3 lead.
Wilkinson did his best to spark the game into life in the second half and a couple of weaving runs worked as the Falcons went up a gear.
Tom May carved his way through and John Rudd crashed over in the corner for Wilkinson to convert from the touchline.