Johnson ran on to the pitch to kick the ball away from Swansea's Izzy Iriekpen after the referee had blown for a free-kick to the West Country side.
This was the second goalless meeting between these Bristol Channel rivals this season.
Swansea needed a win to feed their play-off hopes, while a point was enough for second-placed Bristol to open a five-point gap between themselves and Nottingham Forest.
The Swans had come so close to taking the lead after just three minutes when Dennis Lawrence thundered a shot from the edge of the box against Adriano Basso's right post.
The chance was created by Lee Trundle, who took out three defenders on the right before picking out Lawrence.
A competitive game quickly shaped up into a battle for supremacy between Trundle and the sharpness of home centre-back Louis Carey, once the referee had to come between them with a cautionary word.
At the other end the duel between Iriekpen and Bristol's Stephen Brooker came to a premature end when the striker limped off after half an hour with a recurrence of his right thigh injury.
The Swansea man had been well on top, with Brooker restricted to one clear shot which he lifted over.
Roberto Martinez, the Swans manager had opted for a more orthodox 4-4-2 line-up after his recent experiments. Eager running by Leon Britton on the right flank provided most of the ammunition for his attack, but after their early near-goal they had to wait until the start of the second half for another close call.
Then Thomas Butler, after beating Bradley Orr wide on the left, sent in a dipping shot which Basso pushed out from under the bar.
There were close calls for both sides ending with sub Richard Duffy, who had replaced the injured Kevin Amankwaah, ensuring a draw when he headed out a vicious hook from Carey.