Luke Chambers fired the only goal two minutes into the second half as Forest maintained their promotion challenge to leapfrog Leeds into fifth place in the table.
Forest, who have won six of their last seven league games, are still unbeaten at the City Ground this season and sit just a point behind an automatic promotion spot with a game in hand on their rivals.
"Efficient would be the best way of describing our performance," said Davies. "It was always going to be a tough game after the Derby game at the weekend, which was draining both mentally and physically.
"We knew Bristol would come and make it tough which they did. They came with two banks of four that sat in and they asked questions of us. We had to be patient and knew they would be a threat on the counter attack with the pace of Jamal Campbell-Ryce and Albert Adomah and then Ivan Sproule who came on late on. They are dangerous players.
"But it's a fantastic three points. I don't think we would have won that game last season and that's a credit to the players. The maturity of this side is getting better, the discipline of the team is improving and it's a good clean sheet and three points."
Source: PA
Source: PA