After weathering an early storm, Avon took the lead through a well-struck Fitch penalty.
Edson responded in similar fashion for Keynsham before Fitch regained the lead with another penalty on 27 minutes.
Avon kept up the pressure and after going close on a number of occasions prop Chris Mutlow found himself in space on the left wing to sprint over for the first try. Fitch added the extras to give Avon a deserved 11-3 lead at half-time.
However, Avon could not put Keynsham away and they scored a converted try under the posts following some sloppy defence seven minutes into the second period.
Avon hooker Ben Hurd crashed over for his 11th try of the season but again Keynsham came back and managed to wheel a scrum close to the Avon line, opening up the space to allow Newman to stroll over close to the posts.
Edson missed the conversion, leaving Avon with a one-point lead.
Skipper Kai Harwood pulled level with Hurd once again in the try-scoring charts with his own 11th of the campaign following a prolonged assault on the Keynsham line.
Fitch failed with the touchline conversion, leaving hope for Keynsham and they threw everything at the home side.
Avon defended manfully until the fourth minute of injury time, when they thought they had managed an excellent turnover in the shadow of their own posts.
They were instead penalised by the referee and the Keynham scrum-half took the tap and scored under the posts.
This time Edson made no mistake with the conversion, giving the away side the lead for the first time in the match.
Avon conclude their season with a trip to Matson.